THE  LIBRARY 

OF 

THE  UNIVERSITY 
OF  CALIFORNIA 

LOS  ANGELES 

GIFT  OF 

John  S.Prell 


WORKS   OF   PROF.   M.   A.   HOWE 

PUBLISHED  BY 

JOHN  WILEY  &  SONS. 


The  Design  of  Simple  Roof-trusses  in  Wood  and  Steel. 

With  an  Introduction  to  the  Elements  of  Graphic 
Statics.  Second  edition,  revised  and  enlarged.  8vo, 
vi  + 159  pages,  87  figures  and  3  folding  plates.  Cloth, 
$2.00. 

Retaining-walls  for  Earth. 

Including  the  Theory  of  Earth-pressure  as  Developed 
from  the  Ellipse  of  Stress.  With  a  Short  Treatise  on 
Foundations.  Illustrated  with  Examples  from  Prac- 
tice. Fourth  edition,  revised  and  enlarged.  lamo, 
cloth,  $1.85. 

A  Treatise  on  Arches. 

Designed  for  the  use  of  Engineers  and  Students  in 
Technical  Schools.  Second  edition,  revised  and  en- 
larged. 8vo,  xxv  -f  369  pages,  74  figures.  Cloth,  $4.00. 


Symmetrical  Masonry  Arches. 

Including  Natural  Stone,  Plain-concrete,  and  Rein- 
forced-concrete  Arches.  For  the  use  of  Technical 
Schools,  Engineers,  and  Computers  in  Designing 
Arches  according  to  the  Elastic  Theory.  8vo,  x  + 170 
pages,  many  illustrations.  Cloth,  $2.50. 


A  TREATISE   ON   ARCHES. 


DESIGNED   FOR    THE    USE   OF  ENGINEERS 

AND   STUDENTS  IN   TECHNICAL 

SCHOOLS. 


MALVERD    A.    HOWE,   C.E., 

Professor  of  Civil  Engineering,  Rose  Polytechnic  Institutes 
Member  of  American  Society  of  Civil  Engineers. 


SECOND    EDITION,  REVISED   AND   ENLARGED. 
SECOND    THOUSAND. 

JOHN  S.  PR,_L 

Gvil  &  Mechanical  Engineer. 

SAN  FBANCISCO,  CAL. 

NEW    YORK: 

JOHN    WILEY    &    SONS. 

LONDON:    CHAPMAN   &   HALL,   LIMITED. 

1911 

JOHJM  S.  PRELL 

Qvil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAU 


Copyright,  1897,  1906, 

BY 

MALVERD  A.   HOWE. 


THE  SCIENTIFIC   PRESS 
IERT  DHUMMONO 
BROOKLVN, 


Library 


PREFACE. 


THE  theory  of  the  elastic  arch  as  developed  in  the  follow- 
ing pages  is  based  upon  four  fundamental  equations  demon- 
strated by  Weyrauch  in  1879.  From  these  equations  have 
been  deduced  formulas  similar  to  those  commonly  given  in 
American  text-books,  but  in  a  simplified  form  for  practical 
use.  In  addition  to  these  a  large  number  of  general  formulas 
have  been  introduced,  many  of  which  are  new. 

In  Chapter  V  an  attempt  has  been  made  to  give  a  set  of 
general  formulas  which  can  be  applied  to  any  symmetrical 
arch  either  fixed  or  hinged,  and  subjected  to  either  vertical  or 
horizontal  loads.  These  formulas  readily  reduce  to  the  com- 
mon forms,  and  can  be  applied  in  their  integral  form  to  any 
symmetrical  arch  when  the  equation  of  the  axis  and  the  law 
of  variation  of  the  moments  of  inertia  of  the  cross-sections 
are  known.  In  many  cases  the  reduction  of  the  integrals 
to  a  simple  form  for  a  given  case  would  be  complicated  and 
perhaps  impossible ;  for  such  cases  these  formulas  are  given 
in  their  summation  form  when  they  apply  to  any  symmetrical 
arch  subjected  to  any  loading. 

The  effect  of  the  axial  stress,  which  is  usually  neglected 
by  American  authors,  is  thoroughly  discussed,  exact  as  well 
as  approximate  formulas  being  given  for  all  cases  likely  to 

iii 

737388 

£ngineering 
Library 


iv  PREFA  CE. 

occur  in  practice.  It  is  shown  that  in  flat  arches  fixed  at  the 
ends  the  effect  of  this  stress  should  not  be  neglected  if 
economy  of  material  is  considered. 

Formulas  for  vertical  and  horizontal  loads  are  deduced  for 
each  case  considered,  making  it  possible  easily  to  treat  loads 
making  any  angle  with  the  axis  of  the  arch.  The  effect  of  a 
couple  is  discussed,  and  general  as  well  as  special  formulas 
given. 

Changes  of  temperature  and  of  shape  have  been  con- 
sidered, and  when  not  too  complicated,  formulas  for  special 
cases  are  given. 

Masonry  arches  are  considered,  and  the  many  difficulties 
and  inaccuracies  of  the  common  methods  of  treatment 
pointed  out.  With  a  little  good  judgment  it  is  easy  to  so 
design  a  masonry  arch  that  the  stresses  will  practically 
follow  the  laws  demonstrated  for  the  elastic  arch.  This  has 
been  experimentally  shown  by  the  "  Austrian  Experiments" 
and  by  many  large  arches  designed  and  erected  by  European 
engineers. 

Alexander  and  Thomson's  method  for  designing  seg- 
mental  masonry  arches  has  been  given  as  being  the  most 
consistent  of  the  many  methods  which  assume  all  loading 
due  to  material  to  act  as  vertical  forces  upon  the  arch. 

It  is  hoped  that  the  practising  engineer,  who  has,  as  a 
rule,  little  time  to  study  mathematical  demonstrations  or  to 
search  through  several  pages  of  transformations  for  a  desired 
formula,  will  appreciate  the  collection  in  simple  form  (Chapter 
II)  of  all  of  the  necessary  formulas  likely  to  be  needed  in 
practice,  and  also  the  ease  and  celerity  with  which  they 
can  be  applied,  with  the  aid  of  the  tables,  to  the  case  in  hand. 
A  fair  trial  of  the  summation  formulas  given  in  the  same 
chapter  will,  it  is  believed,  lead  to  the  adoption  of  metal 
arches  more  artistic  in  form  than  the  usual  American  type. 


PREFA  CE.  V 

These  summation  formulas  are  readily  applied  in  the  design- 
ing of  masonry  arches. 

Nearly  all  of  the  formulas  given  have  been  deduced  for 
this  treatise  by  two  radically  different  methods.  Many  of 
these  formulas  are  old,  and  while  it  was  desired  to  give  full 
credit  in  every  particular,  it  was  not  found  either  expedient  or 
possible  to  do  so  for  each  form. 

The  tables  were  carefully  computed,  and  when  possible 
by  the  method  of  differences,  each  tenth  value  being  checked 
by  direct  computation. 

The  demonstrations  are  believed  to  be  sufficiently  simple 
to  be  easily  followed  by  senior  students  in  Technical  schools. 
With  the  aid  of  the  tables,  class  problems  can  be  solved 
which  otherwise  would  be  impossible  on  account  of  the  time 
required  where  direct  computation  of  the  various  terms  must 
be  resorted  to. 

The  author  will  esteem  it  a  favor  if  any  errors  that  may 
be  found  are  at  once  brought  to  his  notice. 

M.  A.  H. 
TERRE  HAUTE,  May  1897. 


NOTE. 


IN  this  second  and  enlarged  edition  the  errors  in  the  for- 
mulas and  example  which  were  discovered  in  the  first  edition 
have  been  corrected,  and  it  is  believed  that  very  few  if  any 
errors  of  importance  remain.  Three  appendices  have  been 
added  which  consider  the  summation  formulas  in  a  simplified 
form  and  also  the  summation  formulas  as  applied  to  unsym- 
metrical  arches.  The  tables  of  arch  data  have  been  rearranged 
and  brought  up  to  date,  and  in  addition  one  reference  has  been 
made,  for  each  item,  to  a  publication  where  a  more  complete 
description  may  be  found. 

M.  A.  H. 

TERSE  HAUTE,  IND.,  July,  1906. 


NOTE   TO  THE   SECOND   THOUSAND    OF   THE 
SECOND   EDITION. 

IN  this  edition  Appendix  I  has  been  rewritten,  and  Table 
XXX  brought  up  to  date. 

M.  A.  H 

TERRE  HAUTE,  IND.,  June,  1911. 

vi 


JOHfl  S.  PRELL 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 
TABLE  OF  CONTENTS. 


CHAPTER   I. 

GENERAL  PRELIMINARY  FORMULAS. 

PAGB 

Deformation  Formulas — Axial-stress  Terms — Distribution  of  Stress  upon 
any  Radial  Section  of  the  Elastic  Arch — Extreme  Fibre-stresses — 
Distribution  of  Stress  when  Arch  has  Two  Flanges  connected  by  Web- 
bracing — Location  of  Resultant  Pressure  for  Like  Stresses  over 
Entire  Section — General  Relations  between  the  External  Forces — 
Equilibrium  Polygons  for  Vertical  and  Horizontal  Loads I 

CHAPTER  II. 
FORMULAS  FOR  PRACTICAL  USE. 

Symmetrical  Parabolic  Arches  -with  Two  Hinges — Vertical  Loads  with 
Effect  of  Axial  Stress  neglected — Vertical  Loads  with  Effect  of  Axial 
Stress  included — Horizontal  Loads  with  Effect  of  Axial  Stress  neglected 
— Horizontal  Loads  with  Effect  of  Axial  Stress  included — Temperature 

— Change  of  Length  in  Span. Symmetrical  Parabolic  Arches -without 

Hinges — Vertical  Loads  with  Effect  of  Axial  Stress  neglected — Vertical 
Loads  with  Effect  of  Axial  Stress  included— Horizontal  Loads  with 
Effect  of  Axial  Stress  neglected — Horizontal  Loads  with  Effect  of  Axial 
Stress  included — Temperature — Change  of  Length  in  Span.- Sym- 
metrical Circular  Arches  with  Two  Hinges — Vertical  Loads  with  Effect 
of  Axial  Stress  included— Vertical  Loads  with  Effect  of  Axial  Stress 
neglected — Horizontal  Loads  with  Effect  of  Axial  Stress  neglected — 
Horizontal  Loads  with  Effect  of  Axial  Stress  included — Temperature 

— Change  of  Length  in  Span. Symmetrical  Circular  Arches  without 

Hinges — Vertical  Loads  with  Effect  of  Axial  Stress  neglected — Hori- 
zontal Loads  with  Effect  of  Axial  Stress  neglected — Temperature  with 

Effect   of   Axial  Stress   neglected. Summation  Formulas  for  Sym* 

vii 


Vlll  TABLE   OF  CONTENTS. 

PACK 

metrical  Arches  of   any    Regular    Shape   and    Cross-section — Vertical 
Loads — Horizontal  Loads — Temperature — Effect  of  Axial  Stress. .  . . . .     2O 


CHAPTER   III. 

PARABOLIC  ARCHES  WITH  THE  MOMENTS  OF  INERTIA   VARYING 
ACCORDING  TO  THE  RELA  TION  Ed  COS  <f>  =  A  CONSTANT. 

General  Relations — General  Formulas  for  Symmetrical  Arches — Symmetri- 
cal Arch  with  Two  Hinges — Vertical  Loads — Change  of  Shape  due  to 
the  Action  of  Vertical  Loads — Horizontal  Loads — Change  of  Shape  due 
to  Horizontal  Loads — Temperature — Change  of  Length  in  Span — Uni- 
form Loads  —  Sinking  of  Supports.  Symmetrical  Arch  -without 

Hinges — Vertical  Loads — Change  of  Shape  due  to  Vertical  Loads — 
Horizontal  Loads — Change  of  Shape  due  to  Horizontal  Loads — Tem- 
perature— Change  of  Length  in  Span— Sinking  of  Supports— Uniform 
Loads. Formulas  for  HI,  MI,  Vi ,  x» ,  xi ,  xt , y<> , y\ , y* ,  etc 52 


CHAPTER   IV. 


General  Relations — Symmetrical  Arches — Symmetrical  Arch  with  Two 
Hinges — Vertical  Loads — Change  of  Shape  due  to  Vertical  Loads — 
Horizontal  Loads — Temperature — Change  in  Length  of  Span — Sinking 

of  Supports. Symmetrical  Arch  without  Hinges — Vertical  Loads — 

Horizontal  Loads — Temperature — Effect  of  Axial  Stress.        Formulas 
Pi,  .to  ,  etc 


CHAPTER  V. 

SYMMETRICAL  ARCHES  HAVING  A  VARIABLE  MOMENT  OF  INERTIA. 

Symmetrical  Arch  without  Hinges — Demonstration  of  General  Formulas  for 

MI   and  HI — Vertical   Loads — Horizontal   Loads — Temperature. 

Symmetrical  Arch  with  Two  Hinges — Demonstration  of  General  For- 
mulas for  Hi — Vertical  Loads — Horizontal  Loads — Temperature. 

Summation  Formulas  for  Arches  with  and  without  Hinges — Vertical 

Loads— Horizontal  Loads — Temperature. Symmetrical  Arch  with  a 

Hinge  at  the  Crown — Vertical  Loads — Horizontal  Loads — Temperature 

—Parabolic  Arch  with  a  Hinge  at  the   Crown. Symmetrical  Arch 

with  Three  Hinges— Vertical  Loads— Horizontal  Loads no 


TABLE   OF  CONTENTS.  ix 

CHAPTER   VI. 

COMPARISON  OF  FOUR  TYPES  OF  ARCHES. 

PAGE 

Comparison  of  the  Values  of  Hi,  V\ ,  Mi,  etc.,  for  Four  Types  of  the 
Parabolic  Arch — Relative  Values  of  Vx  and  Mx  for  Symmetrical 
Parabolic  Arches  with  and  without  Hinges— Comparison  of  Tempera- 
ture Effects  upon  Four  Types  of  Symmetrical  Parabolic  Arches — 
Comparison  of  Maximum  Stresses  for  Three  Types  of  Parabolic  Arches 
—Comparison  of  Weights 145 

CHAPTER  VII.    ) 

APPLICATIONS. 

Point  of  Application  of  Vertical  and  Horizontal  Loads — Wind  Loads- 
Maximum  Stresses— Character  of  Reactions— Co-ordinates  of  the 
Equilibrium  Polygon  —  Bending-moments  at  Supports  —  Series  of 
Examples  illustrating  the  Applications  of  Formulas  given  in  Chapter  II 
—Effect  of  Axial  Stress 159 

CHAPTER  VIII. 

APPLICATION  OF  GENERAL  SUMMATION  FORMULAS  TO  ARCHES  HAVING 
A  HINGE  AT  EACH  SUPPORT. 

Bridge  over  the  Douro  in  Portugal— Data— Computation  of  HI  for  Vertical 
Loads — Comparison  of  Values  of  H\  with  those  obtained  by  Seyrig — 
Values  of  H\  tor  Several  Distributions  of  Moving  Loads — Stress  Di- 
agram for  Load  over  all 182 


CHAPTER  IX. 

APPLICA  TION  OF  GENERAL  SUMMA  TION  FORMULAS  TO  ARCHES 
WITHOUT  HINGES. 

Parabolic  Arch  with  0  cos  <p  =  a  Constant — Data — Computation  of  H\  for 
Vertical  Loads,  using  Summation  Formulas — Comparison  of  H\  from 
Application  of  Summation  Formulas  and  the  Common  Formula — 
Computation  of  M\  for  Vertical  Loads,  using  Summation  Formulas — 
Comparison  of  Results — Computation  of  H\  for  Horizontal  Loads, 
using  Summation  Formulas — Comparison  of  Results — Computation  of 
Mi  for  Horizontal  Loads— Comparison  of  Results— Graphical  Compar- 
ison of  the  values  of  H\  and  MI 190 


TABLE    OF  CONTENTS. 


CHAPTER  X. 
THE  ST.  LOUIS  ARCH. 

PAGE 

Data — Computation  of  H^  and  MI  for  Concentrated  Loads  and  for  Partial 
Uniform  Loads — Comparison  of  Results  with  those  given  in  History 
of  Bridge — Effect  of  Axial  Stress — Temperature — Graphical  Compari- 
son of  the  values  of  HI  and  MI 204 


CHAPTER  XL 
THE  SPANDREL- BRACED  ARCH. 

Douro  Spandrel-braced  Arch — Data — Computation  of  H\  for  Vertical 
Loads,  using  Summation  Formulas — Comparison  of  the  Values  of  H\ 
with  those  obtained  by  Mr.  Max  Am  Ende 217 

CHAPTER   XII. 

THE  MA SONR Y  ARCH. 

Arches  which  can  be  considered  as  Elastic— Thickness  of  Arch-ring— 
Equilibrium  Polygon  following  Axis  of  Arch — Moving  Loads — 
Concrete  and  Brick  Arches — Lead  Joints — Steel  Hinges — Earth-filled 
Spandrels — Masonry  Spandrels 223 


CHAPTER  XIII. 

ALEXANDER  AND  THOMSON'S  METHOD  FOR  DESIGNING  SEGMENTAL 
MASONRY  ARCHES. 

The  Common  Catenary — Transformed  Catenary — Two-nosed  Catenary — 
The  Described  Circle— The  Three-point  Circle— Relative  Positions  of 
Described  and  Three-point  Circles — Horizontal  Thrust — Intensity  of 
Pressure — Advantages  of  Method — Unsymmetrical  Loading. 234 


CHAPTER  XIV. 

EXAMPLES  ILLUSTRA  TING  ALEXANDER  AND  THOMSON'S  METHOD  FOR 
DESIGNING  SEGMENTAL  MASONRY  ARCHES. 

Complete  Solution  of  Several  Problems  showing  Application  of  Tables 247 


TABLE    OF  CONTENTS. 


CHAPTER   XV. 
TESTS  OF  ARCHES. 


XI 


PACK 

Austrian  Society  of  Engineers  and  Architects'  Publication — Results  of  Tests 
of  Five  Full-size  Arches  having  Spans  of  about  Seventy-five  Feet — 
Measurements  of  Deformation — Comparison  with  Theory — Conclusions 
drawn  from  Experiments — Specifications  recommended — Tests  of  Small 
Arches  :  Monier  Arch  ;  Concrete  Arch — Tests  of  Floor-arches — Con- 
clusions drawn  from  Tests  of  Floor-arches 253 


APPENDICES. 

A.  Integrals  employed  in  the  Deduction  of  Ax  for  Parabolic  Arches 263 

B.  Integrals  employed  in  the  Deduction  of  Ay  for  Parabolic  Arches 269 

C.  Effect  of  the  Axial  Stress — Solution  of  Several  Problems  illustrating  the 

Effect  of   the   Axial  Stress — Approximate   Formulas  for   H\  which 
include  the  Effect  of  the  Axial  Stress 272 

D.  Special  Case  :  Semicircular  Arch — Arch  with  Fixed  Ends — Arch  with 

Two  Hinges 284 

E.  Deduction  of  Formulas  for  Special  Cases  of  Chapters  III  and  IV  from 

the  General  Formulas  of  Chapter  V 289 

F.  Effect  of   a   Couple   upon   a   Symmetrical   Arch — Arches    Fixed — The 

Parabolic  Arch 300 

G.  Special  Case  where  the  Moment  of  Inertia  is  Constant — Parabolic  Arch 

with  a  Hinge  at  each  Support 307 

H.  Symmetrical  Arches  having  a  Variable  Moment  of  Inertia — Summation 

Formulas , 309 

I.     Unsymmetrical  Arches  without  Hinges — Summation  Formulas 317 

J.     Unsymmetrical  Arches  with  Two  Hinges,  One  at  Each  Support — Summa- 
tion Formulas 321 


TABLES. 

A,  B,  and  BI  for  Masonry  Arches 326 

I-XVI  inclusive  contain  Functions  used  in  the  Solution  of  Formulas  for 

Parabolic  Arches 330 

XVII-XXIX  inclusive  contain  Functions  used  in  the  Solution  of  Formulas 

for  Circular  Arches 339 

XXX.  General  Dimensions  of  Masonry  Arches 352 

XXXI.  Dimensions  of  a  few  Cast-iron  Arches 357 

XXXII.  Dimensions  of  a  few  Wrought-iron  or  Steel  Arches 338 

XXXIII.  Dimensions  of  a  few  Wrought-iron  or  Steel  Roof-trusses 359 


INTRODUCTION. 


AN  arch  is  a  structure  which,  under  the  action  of  vertical 
forces,  produces  or  exerts  horizontal  or  inclined  forces  against 
its  supports — a  conception  which  does  not  generally  obtain 
outside  the  engineering  profession. 

The  oldest  arch  of  which  we  have  authentic  record  was  dis- 
covered between  1893  and  1896  in  Babylonia.  It  had  a  span 
of  twenty  inches  and  a  rise  of  thirteen  inches,  and,  according  to 
account,  it  was  a  true  ellipse  in  form.  It  was  constructed  of 
well-baked  plano-convex  bricks  laid  as  voussoirs.  The  joints 
were  wedge-shaped  and  made  of  clay  mortar.  The  date  of  the 
construction  of  this  arch  is  placed  4000  B.C.* 

Probably  the  Chinese  first  employed  the  arch  in  the  con- 
struction of  bridges  across  small  streams.  No  authentic  infor- 
mation is  obtainable  in  reference  to  the  time  of  its  first  use. 
It  is  known,  however,  that  bridges  and  other  public  works  were 
executed  in  China  2900  B.C.,  and  that  possibly  the  arch  may 
have  been  used  at  as  early  a  date  as  this. 

Stone  and  brick  arches  have  been  found  in  Egypt,  but  the 
dates  of  their  construction  are  not  positively  known. 

In  "  Campbell's  Tomb  "  an  arch  of  brick  composed  of  four 
ring  courses,  the  inner  ring  having  a  span  of  eleven  feet,  was 
found.  This  arch,  according  to  Wilkinson,  was  built  about 
1540  B.C. 

As  a  rule  the  Egyptians  did  not  use  the  arch  in  their  struc- 

*  "Explorations  in  Bible  Lands  during  the  Nineteenth  Century,"  Hilprecht. 

xiii 


XIV  INTRODUCTION. 

tures,  preferring  a  solid  lintel  as  a  covering  for  openings,  rooms, 
etc. 

A  large  number  of  apparent  arches  have  been  found,  com- 
posed of  masonry  in  horizontal  layers,  corbelled  out  over  the 
openings  and  then  cut  to  resemble  arches.  This  method  of 
spanning  openings  seems  to  have  been  almost  universal,  judg- 
ing from  ruins  found  in  all  parts  of  the  world.  The  Greeks 
employed  this  method  for  covering  quite  large  areas,  although 
it  is  claimed  that  they  were  familiar  with  the  true  arch. 

Here  and  there  throughout  the  Bible  lands  crude  arches  of 
brick  have  been  found.  Beneath  the  palaces  of  Nimrod,  the 
ancient  Calah,  founded  1300  B.C.,  sewers  were  found  covered 
with  pointed  arches  of  brick.  These  arches,  contrary  to  the 
usual  form  of  to-day,  were  inclined  and  could  have  been  con- 
structed without  forms. 

Not  until  about  722  B.C.  have  we  any  record  of  voussoirs 
cut  out  of  stone.  The  gates  to  an  ancient  city  in  Assyria,  now 
represented  by  the  ruins  of  Khorsabad,  were  arched  with  semi- 
circular voussoir  arches  of  stone  having  spans  of  from  twelve 
to  fifteen  feet.  These  are  supposed  to  date  as  early  as  the 
time  of  Sargon,  who  founded  the  city  722-705  B.C. 

To  the  Romans  belongs  the  credit  of  first  using  the  voussoir 
arch  for  spanning  openings  of  considerable  magnitude. 

The  earliest  Roman  arch  of  which  we  have  authentic  knowl- 
edge is  the  Cloaca  Maxima,  constructed  about  615  B.C.  This 
arch  consisted  of  three  concentric  rings  of  stone,  the  inner  ring 
having  a  span  of  about  fourteen  feet.  Like  the  majority  of 
early  Roman  arches,  it  is  semicircular. 

Bridges  of  from  fifty  to  seventy  feet  in  span  were  built  of 
stone  by  .-Emilius  Scaurus,  120  B.C. 

Trajan  (about  104  A.D.)  is  credited  with  having  constructed 
a  wooden  arch  bridge  having  a  span  of  one  hundred  and  seventy 
feet,  but  some  authorities  doubt  that  such  a  structure  ever 
existed. 

One  of  the  largest  stone  arch  bridges  constructed  by  the 
Romans  was  built  by  Trajan,  105  A.D.,  at  Alcantara  in  Spain. 


IN  TROD  UCTION.  XV 

The  largest  arch  was  semicircular  and  had  a  span  of  one  hun- 
dred and  ten  feet.  This  is  probably  the  oldest  stone  arch  bridge 
of  magnitude  which  exists  at  the  present  time. 

Many  aqueducts  were  constructed  by  the  Romans  which 
were  carried  across  valleys  upon  arch  bridges  of  stone,  some- 
times built  in  three  tiers  one  above  the  other. 

Accurate  data  exist  of  many  masonry  arches  constructed 
in  the  seventeenth  and  eighteenth  centuries  by  the  French  and 
the  English,  of  which  the  general  dimensions  of  some  of  the 
largest  and  most  noted  are  given  in  Table  XXX,  which  also 
contains  data  in  reference  to  masonry  arches  built  later. 

The  greatest  distance  spanned  by  a  single  stone  arch  is 
two  hundred  and  ninety-two  feet,  which  is  the  span  of  a  highway 
bridge  completed  at  Plauen,  Saxony  in  1905. 

Concrete  and  reinforced  concrete  arch  bridges  are  now  being 
constructed  in  large  numbers. 

One  of  the  first  large  concrete  bridges  was  built  in  France 
in  1869,  the  longest  span  being  one  hundred  and  sixteen  feet. 
There  has  been  very  recently  completed  in  Ulm,  Germany,  a 
bridge  having  an  arch  span  of  one  hundred  and  eighty-seven  feet, 
while  the  clear  span  between  foundations  is  about  two  hundred 
and  fifteen  feet.  The  ring  has  three  hinges. 

The  largest  reinforced-concrete  arch  span  is  that  of  the 
Griienwald  bridge,  at  Munich,  Bavaria,  built  in  1904,  which 
has  two  spans  of  two  hundred  and  thirty  feet  each.  The  arch 
rings  have  three  hinges. 

Cast-iron  arch  bridges  were  first  constructed  in  England, 
and  the  Coalbrookdale  bridge,  with  a  span  of  one  hundred  feet 
and  a  rise  of  fifty  feet,  has  the  distinction  of  being  the  first  cast- 
iron  arch  bridge  which  was  successfully  constructed.  This 
bridge  was  built  by  Abraham  Darby,  an  iron-founder,  in  1779, 
and  was  in  use  until  1905,  when  it  was  replaced  by  a  lattice 
girder. 

From  this  time  up  to  the  introduction  ol  wrought  iron 
many  very  artistic  cast-iron  bridges  were  constructed;  and 
even  as  late  as  1871  a  cast-iron  arch  bridge  ol  one  hundred 


XVI  IN  TRODUC  TION. 

feet  span  was  constructed  at  Nottingham,  England.  In  the 
United  States  there  are  but  two  cast-iron  arch  bridges  of  any 
magnitude.  One,  the  Chestnut  Street  bridge,  Philadelphia, 
and  the  other  the  aqueduct  bridge,  Washington,  D.  C.,  in  which 
the  arch  ribs  are  composed  of  cast-iron  water-pipe  forty-eight 
inches  in  diameter  and  having  a  span  of  two  hundred  feet. 

The  maximum  span  of  any  cast-iron  arch  bridge  is  that  of 
the  Southwark  bridge,  built  in  1819;  this  has  a  span  of  two 
hundred  and  forty  feet. 

The  dimensions  of  a  few  cast-iron  bridges  are  given  in 
Table  XXXI.  With  but  few  exceptions,  these  bridges  were 
arches  without  hinges. 

The  use  of  wrought  iron  and  steel  in  the  construction  of  arch 
bridges  is  of  recent  date. 

The  first  arch  bridge  *  with  ribs  practically  of  wrought  iron 
was  probably  the  Cron  bridge  at  St.  Denis,  which  was  con- 
structed in  1808.  Wrought  iron  and  steel  have  come  into 
general  use  for  large  arch  bridges  since  1870. 

The  maximum  span  at  the  present  time  is  that  of  the 
Clifton-Niagara  highway  and  trolley  bridge,  which  is  eight 
hundred  and  forty  feet,  centre  to  centre  of  the  end  hinges  or 
pins. 

The  dimensions  of  a  few  wrought-iron  and  steel  arches  are 
given  in  Table  XXXII. 

Wooden  arches  are  probably  not  very  recent.  The  maxi- 
mum span  constructed  was  built  by  Louis  Wernway  in  1812  in 
Philadelphia.  The  bridge  crossed  the  Schuylkill  River,  and 
had  a  span  of  about  three  hundred  and  forty  feet.  It  was  burned 
in  1838. 

Since  the  time  of  the  Romans  the  arch  in  some  form  has 
been  the  favorite  method  for  roof  construction,  in  stone,  wood, 
and  metal,  where  artistic  interior  effects  were  sought  and  means 
were  obtainable  for  executing  the  work. 


*  William   H.   Wahl,  A.M.,   Ph.D.       Iconographic   Encyclopaedia,"   vol. 
p   268 


IN  TROD  UCTION.  XVH 

In  the  United  States  the  arch  is  freely  used  for  roofs  cover- 
ing large  areas,  as  train-sheds,  armories,  exhibition  buildings, 
etc.  These  arches  are  usually  of  metal  and  the  three-hinged 
type. 

The  dimensions  of  a  few  large  roof-arches  are  given  in 
Table  XXXIII. 

Whether  the  ancients  had  any  knowledge  of  the  theoretical 
principles  of  the  arch  is  not  known,  but  it  is  known  that  they 
were  very  successful  in  designing  arch  structures  which  have 
remained  until  the  present  time.  It  is  probable  that  their 
knowledge  was  purely  the  result  of  experiments,  and  in  the 
case  of  masonry  arches  very  little  advancement  has  been  made 
even  up  to  1895,  as  one  may  see  by  comparing  the  dimensions 
and  details  of  arches  constructed  since  1750.  Within  the  past 
ten  years  some  advancement  has  been  made  and  the  arch  rings 
designed  according  to  the  elastic  theory. 

Since  the  time  of  Newton  (1642-1727)  volumes  have  been 
written  upon  the  theory  of  arches,  especially  the  masonry 
arch.  The  theory  of  the  masonry  arch  has  been  and  is  now 
unsatisfactory  from  a  practical  point  of  view,  since  we  are 
unable  to  determine  the  directions  and  magnitudes  of  the  forces 
caused  by  the  materials  above  the  arch  ring  in  the  usual  form 
of  construction.* 

The  necessary  assumptions  which  must  be  made  for  com- 
putation according  to  ordinary  methods  have  been  a  source  of 
much  controversy  among  engineers,  and  will  probably  remain 
more  or  less  of  a  stumbling-block  for  a  long  time. 

The  theory  of  the  elastic  arch  and  its  application  to  metal 
arch  ribs  has  been  developed  since  1840,  and  is  now  generally 
accepted  as  being  sufficiently  accurate  tor  practical  purposes. 
This  theory  is  also  being  accepted  as  the  most  lational  ot  all 
for  the  design  of  masonry  arches,  and  particularly  arches  com- 
posed of  reinforced  concrete. 

*  See  "Symmetrical  Masonry  Arches,"  by  Malverd  A.  Howe  (John  Wiley  & 
Sons),  for  a  discussion  of  this  subject 


NOMENCLATURE. 


NOMENCLATURE  USED   IN  CHAPTERS   II    TO   XI    INCLUSIVE. 
A  =  EO  cos  0  =  constant  for  Parabolic  Arches. 

A  =  — =-  =  constant  for  Circular  Arches. 
jf\ 

a  =  the  abscissa  of  the  point  of  application  of  any  load  P 

or  Q. 

al  and  at  =  the  abscissas  of  the  extreme  limits  of  any  uni- 
form horizontally  distributed  load. 
b  =  the  ordinate  of  the  point  of  application  of  any  load  P 

or  Q. 

c  =  the  difference  in  elevation  of  the  right  and  left  sup- 
ports. 

E  =  the  modulus  of  elasticity. 
e  =  coefficient  of  expansion. 
f=  the  rise  of  the  linear  arch. 
Fx  =  the  area  of  the  arch-rib  at  any  section  x. 
g  —  the  abscissa  of  the  crown  of  the  linear  arch. 
Hx  =  the  horizontal  thrust  at  any  section  x. 
HI  =  the  horizontal  thrust  at  the  left  support. 
//,  =  the  horizontal  thrust  at  the  right  support, 
fj,  =  the  horizontal  thrust  at  the  left  support  due  to  two 

equal  and  symmetrically  placed  loads. 
k  =  a  //. 
k'  =  R  -  f. 
/  =  the  span  of  the  linear  arch. 

/  \*       f) 

m  =  (radius  of  gyration}  =  -^  for  Parabolic  Arches. 


XX  NOMENCLA  TURE, 

I  radius  of  gyration^         Q 

m  =    -~^- =  T^™  for  Circular  Arches. 

\  K  I        rK 

Ml  =  the  moment  at  the  left  support. 
J/,  =  the  moment  at  the  right  support. 
Mx  =  the  moment  at  any  section  x, 
NI  and  N^  =  the  normal  intensities  of   the    resultant  force 

upon  any  section  at  the  extreme  fibres. 
Nx  =  the  normal  component  of  any  force  acting  upon  the 

section  x. 
»=///. 

P  =  any  vertical  concentrated  load. 
p  =  parameter  of  parabola. 
/„  =  the  average  intensity  of  the  resultant  force  acting  upon 

any  section. 

<2  =  any  horizontal  concentrated  load. 
r  =  radius  of  gyration. 
R,  =  the  resultant  of  Vl  and  //,. 
Rt  =  the  resultant  of  F,  and  //,. 
R  =  the  radius  of  a  circular  linear  arch. 
Rx  =  the  resultant  force  acting  upon  any  section. 

s  =  the  length  of  the  linear  arch  curve. 
Tx  =  the  normal  shear  at  any  point  x. 
t°  =  the  number  of  degrees  of  change  in  temperature. 
Vx  =  the  vertical  shear  at  any  section  x. 
PJ  =  the  vertical  reaction  at  the  left  support. 
F",  =  the  vertical  reaction  at  the  right  support. 
w  =  the  load  per  lineal  unit  of  span. 
x  —  the  abscissa  of  any  point  of  the  linear  arch. 
x0  =  the  abscissa  of  the  point  of  intersection  of  Q,  Rt  and  ./?,. 
Xi  =  the  abscissa  of  the  point  where  Rl  cuts  the  horizontal 

passing  through  the  left  support. 
xt  =  the  abscissa  of  the  point  where  R,  cuts  the  horizontal 

passing  through  the  right  support. 
y  =  the  ordinate  of  any  point  having  the  abscissa  x. 
yl  =  the  ordinate  of  the  point  where  Rl  cuts  the  vertical 
through  the  left  support. 


NOMENCLA  TURK.  xxi 

yt  —  the  ordinate  of  the  point  where  Rt  cuts  the  vertical 

through  the  right  support. 

y0  =  the  ordinate  of  the  point  of  intersection  of  Rt  and  Rt. 
a  =  angular  distance  of  the  point  of  application  of  the  load 

P  or  Q  from  the  crown. 

6  —  moment  of  inertia  of  a  normal  section  of  the  arch-rib. 
Bx  =  the  moment  of  inertia  at  the  section  x. 
Q  =  the  angle  made  by  the  resultant  at  any  section  with 

the  horizontal. 
$  =  the  angular  distance  of  any  point  x  from  the  crown  of 

the  arch. 
00  =  the  angular  distance  of  the  left  support  from  the  crown 

of  the  arch. 
<f>{  =  the  angular  distance  of   the  right    support  from  the 

crown. 
AV  Av  etc.,  =  value  of  A,  will  be  found  in  Tables  I,  II,  etc., 

respectively. 

Al  =  small  finite  change  in  /. 
A(f>  =  small  finite  change  in  0. 
A(j)a  =  small  finite  change  in  00. 
A(f>i  =  small  finite  change  in  0/. 
Ax  —  a  finite  value  of  dx. 
Ay  =  a  finite  value  of  dy. 
As  =  a  finite  value  of  ds. 

X 

2  =  algebraic  sum  up  to  the  section  x. 
MASONRY  ARCHES. 

NOMENCLATURE   USED    IN  ALEXANDER  AND   THOMSON'S 
METHOD. 

b  =  distance  of  directrix  to  centre  of  described  circle. 
2c  =  clear  span  of  arch. 

d  =  -distance  of  directrix  from  soffit  of  arch  at  cfowri. '  •  * 
e  =  base  of  Naperian  system  of  logarithms. 
k  =  the  clear  rise  of  arch. 
m  =  the  parameter  of  catenary. 


XXl'i  NOMENCLA  TURE. 

to  =  depth  of  arch-ring  at  the  crown. 
ts  =  depth  of  arch-ring  at  the  skew-backs. 
w  =  weight  of  a  unit  mass  of  masonry. 
r  =  ratio  of  transformation  =  Vs. 
Rl  =  the  radius  of  the  described  circle. 
R,  =  the  radius  of  the  three-point  circle. 
x  and  y  =  general  co-ordinates. 
xl  and  yl  =  co-ordinates     of    the    nose    of    a    two-nosed 

catenary. 

x,  and  7,  =  the  co-ordinates  of  the  point  where  the  two- 
nosed  catenary  is  cut  by  the  three-point  circle. 
y,  =  the  ordinate  of  the  two-nosed  catenary  at  the  crown. 
Ke  =  the  ordinate  of  the  described  circle  at  the  crown. 
*.=Jo-F0. 
<?,  =  departure  of  the  two-nosed  catenary  from  the  described 

circle  at  the  skew-backs,  measured  radially, 
tf,  =  departure  of  the  two-nosed  catenary  from  the  three- 
point  circle  at  the  noses. 

P9  =  radius  of  curvature  of  two-nosed  catenary  at  the  crown. 
pl  =  radius  of  two-nosed  catenary  at  the  nose. 
/oa  =  radius  of  the  two-nosed  catenary  at  the  point  where  it 

cuts  the  three-point  circle. 

0,  =  the  angle  which  pl  makes  with  the  vertical. 
0,  =  the  angle  which  p,  makes  with  the  vertical, 
ytf,  =  the  angle  which  R,  makes  with  the  vertical. 
HI  ,    Vl ,  J/, ,  etc.,  have  the  same  meaning  in  general  as  for 
elastic  arches. 

SOME  FORMULAS  CONSTANTLY  REFERRED  TO. 
Ax  =  -JA^dy  +  cffdx  -  jj  j?-dx (a) 

(b) 


NOMENCLA  TURE.  XX111 


W 


(39) 


F,  =  F,  -     /»   .  .   .  x  >  a  .........  (40) 

Nx  =  Vx  sin  0  +  Hx  cos  0  .........  (42) 

*).  (41) 

.  (49) 

^).  (47) 


(50) 


M,  M, 

^-  and    j,  =  -r  +  r  ........     (51) 


,  a 

=     r     and     ^  =     r  ..........     (54) 


A  TREATISE  ON   ARCHES. 


CHAPTER  I. 

GENERAL  PRELIMINARY  FORMULAS. 
DEFORMATION  FORMULAS.* 

LET  Fig.  i  represent  a  portion  of  an  elastic  arch;  then  the 
relation  between  the  length  of  any  fibre  between  two  adjacent 


radial  sections  can  be  expressed  in  terms  of  the  length  of  the 
neutral  fibre  (limited  by  the  same  radial  sections)  by. the 
equation 

dsn  =  ds  -f-  n  sin  (  —  d<$)  —  ds  —  nd<f>.   .     .     .    (i) 

*  The  formulas  and  demonstrations  in  this  article  are  essentially  the  same  as 
given  by  Prof  Weyrauch  in  "  Theorie  der  Elastigen  Bogentr^ger  "  (Milncben, 
1879).  v 


A    TREATISE   ON  ARCHES. 


Now  suppose  some  circumstance,  as  the  application  of  a 
load,  changes  the  lengths  of  these  fibres,  and  let  s  become 
s  +  As,  sn  become  sn  -f-  Asn  ,  etc.,  as  shown  in  Fig.  2.  Then  we 
have  for  the  new  condition 


Combining  (i)  and  (2), 

dAsH  =  dAs  —  ndA$t 


or 


dAsn 
~dAs 


dAs 


(2) 


(3) 


(4) 


where  dAsM  represents  the  change  in  magnitude  of  dsn  and  dAs 
that  of  ds. 


Let  F'  represent  the  intensity  of  the  force  necessary  to 
change  the  magnitude  of  dsn  by  the  amount  dAsH,  and  let  E 
represent  the  modulus  of  elasticity  of  the  material.  Then 

-^r=2r  •  • (5) 

If  the  force  F'  is  due  to  a  change  of  temperature,  and  e  rep- 


GENERAL   PRELIMINARY  FORMULAS.  3 

resents  the  coefficient  of  expansion  per  degree  of  change,  then 
for  an  increase  of  temperature  of  t°  we  have 


Let  Nn  be  the  intensity  of  any  normal  force  acting  upon  the 
fibre  sn  ;  assuming  that  this  force  acts  at  the  same  time  with 
the  change  of  temperature  but  of  opposite  effect,  then  we  have 
from  (5)  and  (6) 


From  (i)  and  (2), 

dAsn dAs  —  ndAtf) 

dsn   ~       ds  —  nd(f> 

Substituting  (8)  in  (7)  and  solving  for  NH ,  we  obtain 


(8) 


ds- 
or 

~dAs  — 


(9) 

+  Eet°.    .    .    (10) 


H~ds~ 


d(f>  i 

Now  ds  =  R  sin  (  —  defy  =  —  Rd<f>,  or  -r-  =  —  -r,;  hence 

dAs        R 


Let  /'  represent  the  area  of  the  fibre  sn,  and  Nx  the  resultant 
normal  force  acting  upon  the  section.     Then 


4  A    TREATISE   ON  ARCHES. 

If  we  take  the  centre  of  moments  on  the  axis  of  the  arch  at 
the  section  x,  the  moment  of  the  radial  force  upon  the  section 
will  be  zero. 

If  x0  is  the  distance  of  the  point  of  application  of  the  force 
Nx  from  the  axis,  we  have 


=themoment  of  the  external  forces  acting  uponthe  section  ; 
then  from  (12)  we  can  write 

f'n^R      dAsf'nR 


'n..     .     .     (13) 


Let  R2~-     =  Wand  2f  =  Fx  ;  then  2f'n  =  o; 


hence  in  (12) 

f>nR  V  W 


N 
Substituting  these  values  in  (12)  and  solving  ior~t  we  obtain 

W 


or 

Nx  (  dA<f>    .     i  dAs     W 


(13)  reduces  to 

Mx       ('dA<t>n    .  dAs  )  W 


(15) 


GENERAL  PRELIMINARY  FORMULAS. 

From  (15) 

(  dA  0  .     i  dAs  \   Mx 

\~~ds~^~  Tt~ds  \  ~~ EW 

Substituting  this  in  (14), 

N*  -   _  M*  __  dAs 
or 


Substituting  (16)  in  (15)  and  reducing,  we  have 

\N>MX\      i      4_Mx_et°_ 
N*-Jr-  -- 


Substituting  (16)  and  (17)  in  (11),  and  reducing,  we  have 

Mx      Mx     nR 


From  Fig.  2, 

d(x  4-  Ax)  -  d(s  4-  As)  cos  (0  4-  J0) ; 

d(y  4~  Ay)  =  ^  4~  ^)  sl"n  (0  4~  ^0) » 
but 

^          xf^ 

cos  (04-J0)=cos  0  cos  A(b— sin  0  sm  J0  =  — — A<P-f  » 


sin  (0+J0)=sin  0  cos  J0  +cos  0  sin  J0  =  -    +A$—.    (20) 


- 

Substituting  (19)  and  (20)  in  the  two  expressions  above, 
and  integrating,  we  have 


Ax  =  -AQdy  +-±(dx  -  dyA<t>)  ; 
or,  from  (16), 


Ydx  -vAQdy.    .     .    (21) 


A    TREA  TISE   ON  ARCHES. 


But    /  YA(f)dy  can  be  neglected  in  comparison  with  the  terms 
preceding.     Hence 


......    (22) 

In  a  similar  manner, 

(23) 
From  (16)  and  (17), 

,      ...........     (24) 

and 

A$=fxds  ............     (25) 

These  four  equations,  (22),  (23),  (24),  and  (25),  completely 
determine  the  effect  of  any  change  of  position  of  any  point  in 
the  axis  of  the  arch  when  X,  Y,  and  the  equation  of  the  axis 
of  the  arch  are  known. 

The  expressions  for  X,  Y,  and  NH  can  be  simplified  by 
replacing  Wby  6  =  ~2f'n*  =  the  moment  of  inertia  of  the  cross- 

section.    Since  in  the  expression  W  =  R2      "    ,  R  is  usually 

n  -\-  K 

very  large  in  comparison  with  n,  no  material  error  results  from 
the  change. 

In  (16)  is  a  very  small  quantity,  and  consequently 

we  can  neglect  and 


"  (!7)  02  A-  anc*  i^  can  be  omitted. 
K  rLrx  K. 

In  (18)  —  -^  can  be  neglected  and  —  -  —  -  be  assumed  to 
Krx  n  -j-  K 


equal  n. 


GENERAL   PRELIMINARY  FORMULAS.  7 

Making  these  modifications   and  collecting   our  formulas, 
we  have  finally 


(d) 


The  term  containing  Nx  shows  the  effect  of  the  axial  stress, 
which  is  usually  neglected  in  the  common  investigation  of  the 
problem.  In  many  cases  the  influence  of  this  stress  is  of  little 
or  no  importance,  but  in  very  flat  arches  it  should  not  be 
neglected. 

Omitting  the  terms  containing  Nx  greatly  simplifies  the 
deduction  of  equations  for  special  forms  of  arches,  and  also 
the  solution  of  these  equations  in  the  determination  of  the 
reactions,  bending-moments,  etc. 

Omitting  the  terms  containing  Nx,  we  have 

Ax=  —  J*  AQdy  +  effdx  ;  ....  (aa) 
Ay  =f  A<pdx  +  efjdy  ;  .....  (bb) 
.........  (cc) 

*;     .......  (dd] 


A    TREA  TISE   ON  ARCHES. 


THE  DISTRIBUTION  OF  STRESS  UPON  ANY  RADIAL  SECTION 
X  OF  THE  ELASTIC  ARCH. 

In  Fig.  3  let  A  C  represent  any  radial  section  of  the  arch, 
and  Nx  the  resultant  normal  force  applied  to  the  section  at  a 
distance  xa  from  the  axis  of  the  arch  passing  through  the 
centre  of  gravity  of  the  section ;  then 


FIG.  3. 
From  (e),  after  substituting  (29), 


_N       n 

—   -77-     I     -a 


or 


(30) 


Now  ~  represents  the  average  intensity  of  the  pressure 

I'M 

on  the  section,  and  may  be  represented  by/0  for  convenience ; 

N  F  x 
and  — *   *  °n  represents  an  intensity  which  varies  directly  with 

n ;  hence  NH  is  composed  of  the  algebraic  sum  of  an  average 
intensity  and  a  uniformly  varying  intensity. 


GENERAL   PRELIMINARY  FORMULAS.  $ 

Replacing  —  -  by  /„,  and  remembering  that  _?  =   *,  where 
r  represents  the  radius  of  gyration,  (30)  becomes 


from  which  the  intensity  of  pressure  at  any  point  of  the  section 
can  be  determined. 

Let  -  =  k,  and  -  =  k,.     (Fig.  3.) 
a,  a* 


Making  n  =  0,  in  (/),  we  have 


Making  n  =  —  a2  in  (/),  we  have 

x( 
~~k. 


(31) 


(32) 


(31)  and  (32)  determine  the  intensities  upon  the  extreme 
fibres  of  the  section. 

When  x,  =  —  £,,  N,  =  o. 

«      Xn  =  Jrk1,  N,  =  o. 

"      xfi  >  —  &,,  Nl  and  Nx  have  the  same  sign. 

«      ^o<_|_^u  N^  and  Nx     "      "        "         " 

6 


FIG.  4. 

Hence  when  x9  >  —  ^  and    <  ^  the  entire  section  is  sub- 
jected to  the  same  kind  of  stress. 

To  illustrate :  Suppose  the  section  to  be  rectangular ;  then 
F  =  bh  and  &  -  ^bh\  and 


IO  A    TREA  TISE   ON  ARCHES. 

N  =  o  when  I  4-  — -  —  o,    or  x^  -— =  —  k~ ;    or   the 

k  6 

resultant  stress  acting  upon  the  section  must  cut  the  sec- 
tion at  a  point  distant  from  the  axis  one  sixth  the  depth  of 
the  section  and  below  the  axis  (see  Fig.  4).  Evidently,  if  Nx 

were  applied  above  the  axis  and  x^  =  -,  N^  =  o  and  Nl  =  2p0. 

Hence,  in  order  that  all  parts  of  a  rectangular  section  be  sub- 
jected to  the  same  kind  of  stress,  the  resultant  stress  Nx  must 
be  applied  within  the  middle  third  of  the  section. 
Adding  (31)  and  (32), 


&>  +  &,  =  2/0  -i-  A*o(gi-*»). 


If  a,  =  av 


Returning  to  (e), 


From  Fig.  3,  letting  Q  represent  the  force  whose  intensity 
is  uniformly  varying,  we  have 


But  Mx  —  7V>0  =  QAt',  therefore 

e          e 


2f'n 


(35) 


(36) 


which  completely  determines  the  arm  of  the  couple  whose 
moment  is  Q/i0.  Now  since  the  intensities  of  the  force  Q  vary 
directly  with  n,  the  intensity  at  the  axis  of  the  arch  must  be 


GENERAL   PRELIMINARY  FORMULAS. 


II 


zero,  and  the  application  of  Q  be  £//0  from  the  axis  as  indicated 
in  Fig.  3. 

ARCHES   HAVING  TWO   FLANGES   OR   CHORDS. 

In  case  the  arch  is  composed  of  two  flanges  connected  by  a 

thin  web  or  by  struts  and  ties,  it  is  customary  to  consider  the 

material  of  each  flange  concentrated  at  its  center  of  gravity, 

and  that  the  flanges  resist  all  stresses  excepting  radial  stresses 


From  Fig.  5, 


or 


_         . 
xt  — 


(37) 


ft! 


Also 
and 


FIG.  5. 


,,  _  NJi,  -  xnNx  _  h,Nx  -  Mx 
' 


•    (38) 


Q  and  Q"  will  be  of  the  same  kind  as  long  as  -f-  x0  is  less 
than  h^  and  —  x^  less  than  //,,,  or  Nx  must  lie  between  Q  and 


VALUES   OF  X0   FOR  VARIOUS  SECTIONS. 

The  following  table  contains  the  maximum  values  which  x0 
can  have  when  the  entire  cross-section  is  subjected  to  the  same 
kind  of  stress  for  the  various  sections  shown. 


A     TREA  TISE    ON  ARCHES. 


*=*  = 


'.  =  ± 


4 


«,  =  #*  =  \b, 


»=  ± 


a.  =  a.= 


1.414 


t-2py 


F  =  2.598^,        *,  =  *,=  o.866£, 
0  =  0.5413^,       r'  =  0.2083^, 


F  =  2.598^',         a,  =  at  =  b, 
9  =  0.5413^,        r1  =  0.2083^*, 

**  =  ±  0.2083^. 


d!,    =  tf,    =    O.924^, 

0.638^*,         r'  =  0.2256^', 
xa=  ±  0.2446. 


F  =&&-&&, 

e 


h 


12  bk  - 


i  M-  -  bfc 

o  —   -1-  Z7  ~T7 TT~' 

o//  M  —  b.h. 


GENERAL   PRELIMINARY  FORMULAS. 


9 

F-  bh  +  bh                    -  a   -h 
*  -r  .  i»          #t  —  #,  —  2> 

12  £//-[-  ^/^  ' 

"*"  6^  bh  -\-  bji^  ' 

Si 

p  -  bh       (I       b  U                            h 

12  bh  -(b-  b,)h,  ' 

(«-—&—  4 

6/z  bh-(b-  b^/i,  ' 

n 

F=-d*  =  0.7854^',         a,  =  a,=  -, 

6  =  0.049  1  <af4,                     r>  =  0.062  5</f, 
^•0  =  ±  O.I25</  =  ±  i  radius. 

m 

d 

2 

8  =  0.049  1  (d?4  —  O>         r*  =  0.062  5(^'+^,*), 

SB 

F  =  0.78  5  4#£,        ^,  =  «.,  =  -, 
^  =  0.049  1  b/i3,       r2  =  0.062  5  //', 

X0  =    ±  O.\2$k. 

® 

F  =  0.78  54(£/*  —  *//,),    at=at  =  -, 

_,_  0.125^'  —  ^,^13 

-t-0  —  -L      >      /  /       »  / 

ll         Oil  —  0/2, 

A    TREATISE   ON  ARCHES. 


GENERAL   RELATIONS   BETWEEN   THE   EXTERNAL   FORCES. 

In  Fig.  6  let  ABC  represent  the  axis  of  any  elastic  arch, 
and  P  and  Q  the  vertical  and  horizontal  components  respec- 


FIG.  6. 


tively,  of  any  load  applied  at  a  point  having  the  co-ordinates  a 
and  b ;  then  for  equilibrium  we  have 


x  =      l-2-       x>a 
Vx-  V,  -i/7;       x>a 

Mx  =  M,  +  F^  -  Hj  -  ±P&  -  a) 
Referring  to  Fig.  7, 

A7;  =  Vx  sin  0  +  Hx  cos  0  ;    . 


,  =  ^  cos  0  —  //,  sin  0  ;   . 

Vx 
tan   ?  =  - 


(39) 

(40) 


(42) 

(43) 

(44) 


;3.  (45) 


GENERAL  PRELIMINARY  FORMULAS. 

Differentiating  (41), 


tan  0. 


FIG.  7. 
Since  dx  —  ds  cos  0,  we  have  from  (43) 


Also,  from  (40),      Vx  =  V,  -  2P. 
Hence 


=VX-H,  tan  0  +  i"£  tan  0 
«.# 

=  T;      +  Hn  tan  0  -  (^i  -  10  tan  0 


Therefore 


dM, 
ds 


=  T, 


(46) 


Making  x  =  /  and  j  =  £  in  (41),  Af,  becomes  M9t  and  we 
have 


1 6  A    TREATISE    ON  AKCffES. 

Solving  this  for  J7,,  we  obtain 

Making  x  =  I  in  (40),  and  combining  with  (47),  we  have 

Collecting  the  equations  which  will  be  employed   in    the 
investigation  of  special  cases,  we  have 


rM=  V,  -  2P\      x>a.  .     .     .     . 

Nx  —  Vx  sin  0  +  Hx  cos  0.       ... 

Ml  =  Mt—  VJ  +  Hf  +  2P(t  —  a)  — 
V,—  j{  Mt  —  M,  +  H,c  +  2P(l  —  a)  — 


(39) 

(40) 
(42) 

(40 
(49) 

(47) 


-*)}.    •     (48) 


ORDINATES   LOCATING   THE   EQUILIBRIUM    POLYGONS. 

(a)    Vertical  Components. 
In  Fig.  8  let  ABC  represent  the  axis  of  any  elastic  arch 


and  let  a  single  vertical  load  (corresponding  to  the  vertical 
component  of  any  load)  be  applied  at  B.     Thas  load  causes 


GENERAL   PRELIMINARY  FORMULAS.  I/ 

the  reactions  R^  and  Rt  and  the  moments  Ml  and  Mt  at  A  and 
C  respectively.  This  condition  can  be  represented  graphically 
by  the  equilibrium  polygon  GEK,  which  must  be  so  situated 

V  V 

that  Hji  =  M,  ,  H^(y^  —  c)  =  M,,  tan  /?,  =  -j~,  and  tan  /3t  =  -~. 

From  Fig.  8,  taking  moments  about  E, 


or 


which  locates  the  point  E  when  Mlt  Vlt  and  Hl  are  known. 
Taking  moments  about  D, 

M,-H,yl=Q,     or    7,  =  -.      .     .     .     (51) 


Taking  moments  about  F, 

M9-Ht(^-c)  =  o    or    ?t  =  <:+^-..     •    (52) 
From  the  triangles  DGA  and  CKF, 

tan  £  =  5-      and     tan  fr  =  ^-.    .     .     .    (53) 

/7j  •"» 

From  the  triangles  GAD  and  GLE, 


and 


3 (54) 


We  also  have 


x  -  ,    .    .    .    (55) 

•*«  —     77  %**/ 

Ka 

The  above  equations  completely  determine  the  locations 
of  GDEF  and  K,  and  hence  the  equilibrium  polygon  GEK 


i8 


A    TREATISE   ON  ARCHES. 


can  be  drawn  in  its  true  position  and  the  values  of  R1  and  Rt 
at  once  determined. 

Having  determined  Rt  and  R,  in  magnitude  and  position, 
the  distribution  of  pressure  over  the  section  at  A  can  be  found, 
and  then  the  stresses  in  other  portions  of  the  arch  determined. 
When  the  arch  is  solid  in  section  the  stresses  are  best  deter- 
mined by  equations  (39),  (40),  etc.  If,  however,  the  arch  is 
composed  of  two  flanges  connected  by  a  thin  web  or  by 
bracing,  the  graphical  method  is  the  more  expeditious. 

The  methods  of  determining  the  stresses,  etc.,  at  different 
points  of  the  arch  will  be  fully  illustrated  by  problems,  but  a 
brief  outline  of  one  method  of  procedure  after  Rt  has  been 
determined  will  be  given  here. 

In  Fig.  9  let  AB  be  any  radial  section  of  a  solid  elastic 
arch.  Suppose  I,  2,  ...  5  represent  the  positions  and 


FIG.  9. 

magnitudes  of  the  resultants  for  five  vertical  loads.  Then 
the  position  of  their  resultant  and  its  magnitude  can  be  found 
graphically  as  shown.  The  distance  xn  can  now  be  scaled,  the 
force  R  resolved  into  the  components  Nx  and  Tx,  and  the 
stresses  upon  the  section  AB  completely  determined.  (See 
page  8.) 

If  the  arch  is  composed  of  flanges,  the  method  is  practi- 
cally the  same,  with  the  exception  that  each  flange  is  assumed 


GENERAL   PRELIMINARY  FORMULAS.  19 

to  have  a  uniform  stress  over  its  entire  section,  as  explained 
above.     (Seepage  n.) 

(b]  Horizontal  Components. 

An  examination  of  Fig.  10  shows  that  we  can  locate  the 
equilibrium    polygon    GEK  for  the  horizontal   load    Q   in  a 


!- * 

-—-"--4--- 

OJs- •* 


FIG.  10. 


manner  similar  to  that  employed  for  the  vertical  component ; 
in  fact,  all  of  the  equations  will  be  the  same,  with  the  exception 
of  that  for  j0,  which  in  this  case  becomes  x0. 
We  have  then 


M,  M, 

*  and    *•*-<= 


Also, 


*i = 77  and  *•  =  y,- 

From  the  figure 


(56) 
(57) 


or 


(58) 


CHAPTER   II. 
FORMULAS   FOR   PRACTICAL  USE. 

IN  this  chapter  all  of  the  important  formulas  for  parabolic 
and  circular  arches  have  been  collected  and  arranged  for  ready 
application.  The  demonstrations  of  these  formulas  are  given 
in  chapters  which  follow. 

(A)    SYMMETRICAL  PARABOLIC  ARCHES. 
A  =  Ed  cos  <f>  =  a  constant,  or  6  varies  inversely  as  cos  <p. 

EB  cos  (f>  =  A  —  a  constant;        X59*)>  •     •     (59) 

a 

m  =  ~~  —  (radius  of  gyration)1 ;         X^°)     •     (6°) 
r 

I* 

p  —  parameter  of  parabola  =  —  ;  .     .     .     .     (61) 

o/ 

b  =  4/£(i  —  k}f  =  y     for    x  =  a  =  kl\       .     (62) 

f)/. .    (63) 

J,  =  function  given  in  Table  I, 
J,  =  function  given  in  Table  II, 
J,  =        etc.         etc. 

ARCH   WITH   TWO   HINGES,   ONE  AT   EACH   SUPPORT. 

(a)   Vertical  Loads,  with  Effect  of  Axial  Stress  neglected — 
Common  Method. 

H^\l-^P\k(i-2k^k^-\        .     .     p(gi)    .     .     (64) 

o/ 

*/(59)  indicates  that  this  equation  is  taken  from  the  chapter  on  Parabolic 
Arches  (Chap.  Ill),  its  number  in  that  place  being  />(59). 


FORMULAS  FOR  PRACTICAL    USE. 


21 


or 


\-L-kpA  ^A^  =  function  given  in  Table  I).     .  (640) 
°y 


X93)    •     .     (65) 
/(95)    •     •    (66) 


FIG.  ii. 


or 


(66*) 


Tx  =(Vl-  2P)  cos  0  -  H,  sin  0. 
From  (42), 

Nx  =  (V,  -  i\P)  sin  0  +  //;  cos  0. 
From  (41), 


Mx  =  V,x  -Hj- 


-  a). 


(42) 


/(97) 


•     (67) 
.    (68) 

.     (69) 


The  application  of  the  above  formulas  to  either  the  solid 
or  open  arch  rib  is  quite  simple.  The  formulas  are  exact,  of 
course,  for  the  solid  rib  alone,  and  then  only  when  the  depth 
of  the  rib  is  small  and  the  loading  is  applied  upon  the  centre 
line;  yet  for  practical  purposes  they  can  be  applied  to  open 
ribs. 


A    TREA  TISE    ON  ARCHES. 


SOLID   RIB. 

For  the  solid  rib  we  compute  the  values  of  Hl  and  Vlt  and 
then  determine  the  values  of  Mx,  Nx,  and  Tx  for  each  section  of 
the  arch,  the  sections  being  taken  at  convenient  distances  apart. 

The  values  of  Mx,  Tx,  and  Nx  can  be  found  from  a  graphi- 
cal construction  as  shown  in  Fig.  12. 

Draw  the  locus  line  S  after  computing  j0  (formula  66),  and 
then  draw  FA  and  FC  for  the  load  being  considered.  By  reso- 
lution of  forces  R^  R^  //,,  Ht,  Vlt  and  V^  can  be  determined. 


FIG.  12. 

Let  ab  be  any  radial  section  where  Mx,  Nx,  and  Tx are  desired. 
Then  Nx  equals  the  normal  component  of  R^  upon  ab,  Tx  equals 
the  tangential  component  of  ^  upon  ab,  and  Mx  equals  the 
ordinate  (o)  multiplied  by  N* 

Maximum  Value  of  Mx. — Let  ab,  Fig.  14,  be  any  section 
where  the  maximum  moment  is  desired.  Draw  the  lines  Ao  and 
Co  until  they  cut  the  locus  line  S.  Then  since  the  loads  at 
these  points  produce  no  moment  at  the  section  ab,  these  points 
separate  the  loadings  which  cause  moments  of  different  signs. 
The  shaded  sections  in  the  figure  clearly  indicate  the  loadings 
which  cause  maximum  ±  Mx. 

Maximum  Value  of  Tx. — At  any  section  ab,  Fig.  13,  draw 
AD  perpendicular  to  ab.  Then  the  loading  causing  positive 


FORMULAS  FOR   PRACTICAL    USE.  2$ 

and  negative  shear  is  distributed  as  shown  by  the  shaded  por- 
tions in  the  figure ;  or,  for  maximum  upward  shear  the  arch 
must  be  loaded  on  the  left  up  to  the  section  ab  and  on  the  right 
between  D  and  E. 


FIG.  14. 

Uniform  Loading.  —  Thus  far  only  concentrations  have  been 
considered.  If,  however,  the  loading  is  uniformly  distributed 
horizontally,  we  have 


n  —f/l  and  w  =  load  per  unit  length  of  span. 


(70 


24  A    TREATISE   ON  ARCHES. 

V 

_  wlr  (  y 


Tx  =          *(2  -  *)  cos  0  -  -1- 


-  (wl(kf  -  k"}  cos  0,  where  £'<j).      .    /(i35)         (73) 

K a'=fcV * 

_t 


FIG.  15. 
OPEN    RIB. 


The  values  of  Hl  and  F",  are  found  by  the  formulas  given 
for  the  solid  rib.  The  internal  stresses  can  then  be  found  by 
Clerke-Maxwell's  method  of  graphics. 

The  loadings  causing  maximum  values  of  Mx  and  Tx  are 
clearly  defined  in  Figs.  16  to  18  inclusive. 

Fig.  1 8  is  strictly  true  only  when  cd  and  ab  are  parallel. 

PLATE-GIRDER    RIB. 

In  plate-girder  ribs  the  flanges  usually  are  assumed  to  resist 
the  bending-moment,  and  the  web  the  shear  or  Tx\  hence  we 
may  treat  them  the  same  as  the  open  rib. 

EQUILIBRIUM    POLYGON. 

If  in  any  of  the  above  cases  it  is  desired  to  construct  an 
equilibrium  polygon  for  any  given  loading,  it  can  be  done  as 


FORMULAS  FOR  PRACTICAL    USE. 
Piece  :  06 


FIG.  1 8. 


26 


A    TREA  TISE   ON  ARCHES. 


follows  (Fig.  19) :  Construct  the  resultants  R^  for  each  concen- 
tration, and  find  the  resultant  of  the  system,  also  the  corre- 


FIG.  19. 

spending  values  of  //,  and  V^\  then  the  polygon  can  be  con- 
structed by  the  usual  methods. 

(b]   Vertical  Loads,  with  Effect  of  Axial  Stress  included. 

8//V  _      _  ml* 
15 


where  £}  is  the  value  found  from  (640)  for  //lf  or  approxi- 
mately 


(75) 


(Tables  I  and  V.) 

The  axial  stress  affects  only  the  value  of  //i;  hence  to  in- 
clude the  effect  of  the  axial  stress  we  have  merely  to  compute 
Hl  by  (74),  and  then  proceed  as  already  outlined  for  the  case 
where  the  axial  stress  is  neglected. 


*See  Appendix  C. 


FORMULAS  FOR  PRACTICAL    USE. 


If  the  locus  line  S  is  determined  by  computing  the  ordinates 
R   they  must  be  deduced  from  the  formula 


(50)        (7<5) 


where  //,  is  to  be  found  from  (74). 


(c)  Horizontal  Loads,  with  Effect  of  Axial  Stress  neglected  — 
Common  Method. 


/(in)    -    (77) 
....  (77*) 


FIG.  20. 


/(1 13)      •     (78) 


.     .     .(78*) 
;n6)    .    (79) 


From  (39),  (40),  and  (43), 

=  F,  cos  0  -(//,-  i<2)  sin  0 


(43)     .     .     (80) 


28  A    TREATISE   ON  ARCHES, 

From  (39),  (40),  and  (42), 

Nx  =  V,  sin  0  +  (//,  —  i'0  cos  0  .....   (42)     .     .     (81) 
From  (41), 

-     (82) 


The  application  of  the  above  formulas  to  either  the  solid  or 
open  arch  rib  is  quite  simple.  After  the  locus  line  S  has  been 
located  by  means  of  (79),  the  reactions  can  be  drawn  as  shown 
in  Fig.  20,  and  from  these  the  values  of  //"and  V  determined. 

Horizontal  loads  are  usually  caused  by  wind  ;  hence  the 
ordinary  case  to  consider  is  half  the  arch  covered  with  a  steady 
load. 

(d)  Horizontal  Loads,  with  Effect  of  Axial  Stress  included. 
-*i  -k- 


+  20'™^-' A»5)     •    (»3) 

where 

£=8//'  +  Io,»M /(I26)      '     (84) 

•*"•  =  y* = -g7- (58)  .  .  (85) 

The  axial  stress  does  not  affect  the  other  equations  ex- 
cepting where  they  contain  //,  or  xtt  the  values  of  which  must 
be  found  from  the  above  expressions. 

After  the  values  of  Hl ,  Vl,  etc.,  have  been  determined  for 
horizontal  loads,  the  stresses  can  be  found  in  the  manner  out- 
lined for  vertical  loads. 


FORMULAS  FOR  PRACTICAL    USE.  29 

(e)   Temperature. 

H.  =  ^t°,      ....   /(I28)      .     (86) 
neglecting  the  effect  of  the  axial  stress  ;  or 


including  the  effect  of  the  axial  stress. 

A  rise  in  temperature  causes  a  horizontal  thrust  similar  in 
character  to  that  produced  by  vertical  loads  acting  downward. 

V,  =  o  ........   (47)    -     •     (88) 

TM=-ffl  sin  0.     ...   (43)    •     .     (89) 

Mx=-  Hj>  ......   (41)    .     .     (90) 


In  case  of  the  open  rib  arch  the  stresses  in  the  individual 
members  can  be  found  by  graphics  after  Hl  has  been  deter- 
mined. 

(f)   Change  of  Length  in  Span. 

By  replacing  ef  by  —  in  the  above  equations  they  may  be 
applied  to  any  change  in  length  of  span. 


ARCH   WITHOUT   HINGES. 


(a)    Vertical  Loads,  neglecting  Effect  of  Axial  Stress  — 
Common  Method. 


(91) 


or 


r  =  -±2/>JM,  where  n  =///. to**) 

4* 


3° 


A    TREATISE   ON  ARCHES. 


M,  =  - 


reading  (i  -  k)  for  k. 


(92(Z) 


FIG.  21. 


^=2j\i-k)\l+2k) 


or 


or 


(93) 

(93«) 

-/    (positive  upwards) /(ISO       (94) 

^  Q(l-t)S  =  SJ»  reading  (i  -  k)  for  &     /(i53)       (96) 

(97) 

(97*) 


FORMULAS  FOR  PRACTICAL    USE.  3! 

< X.55)    -      (93) 


,  =  -—  J9,  reading  (I-  £)  for  £  ........    (980) 


Tx  =  (  V,  -    V)  cos  0  -  H,  sin  0  .....     (43)    .      (99) 

-  a).     .    .     .    (41)    .     (100) 


As  in  the  case  of  the  two-hinged  arch,  it  is  necessary  only 
to  compute  //",,  F,,  and  ja  to  determine  all  the  outer  forces 
acting  upon  the  arch,  and  then  the  stresses  ;  but  as  a  check  it 
is  advisable  to  compute  x^  xy,  j,  ,  and/,. 

The  methods  of  determining  the  fields  of  loading  which 
cause  maximum  values  of  Mx  and  7  x  are  the  same  as  for  the 
two-hinged  arch,  only  the  resultants  Rt  and  Rt  do  not  neces- 
sarily pass  through  the  supports,  but  must  have  their  locations 
fixed  by  the  ordinates  x^  ,  yl  ,  xtJ  and  y^ 

(b)  Vertical  Loads,  with  Effect  of  Axial  Stress  included. 

H,  =  C\  &Pk\i  -  ^  -  2/(  j^2/)i^(i  -  k)  }  ,    .    (101) 

/(i62) 
where 


Approximately, 
where  ^  =  //t  in  (91). 


*(i-*)  =  4  .......    (105) 

k\i  -  kj  -  Jlt  .......     (106) 


*  See  Appendix  C. 


32  A    TREA  TISE   ON  ARCHES, 

Note  that  all  quantities  in  tJie  above  equations  excepting  those 
given  by  the  tables  are  constant  for  any  given  arch. 


-  lD2Pk(i  -  2k*  + 


.    (107) 


^=I  +  ^_rzpL (I08) 


where  //,  is  to  be  found  from  (101). 


k(i  —  2k?  +  /£')  =  //, (109) 

2k  —  3/P  -f  k3  =  J10.    ......    (no) 

£(i  —  /£)  =  z/6 (in) 

*  To  determine  J/,  replace  k  by  (i  —  £)  in  (107),  or  compute 
ra  from  (107)  and  substitute  the  value  in 


(so 


*  Note   that  only  the  terms  containing  2k  —  •$&  -f-  &  and    zk  —  i   change  in 
magnitude  when  I  —  k  is  used  in  place  of  kc 


FORMULAS  FOR  PRACTICAL    USE. 


33 


Having  computed  Hl ,  Ml ,  Mt ,  F, ,  and  yt ,  all  the  other 
outside  forces  can  be  found  as  follows(Fig.  22) :  Lay  off  H,  and 
F,  at  A  and  complete  the  parallelogram  of  forces,  thereby  de- 
termining the  direction  and  magnitude  of  R^.  Then  lay  off  yl 


FIG.  22. 

at  A  above  or  below,  according  to  the  sign,  and  draw  Rl  in  its 
proper  position,  extending  its  direction  until  it  cuts  the  load 
being  considered.  By  parallelogram  of  forces  F,  and  R^  are 
readily  found.  As  a  check,  //,,  F,,  and  yt  should  be  com- 
puted. 

(c)  Horizontal  Loads,  with  Effect  of  Axial  Stress  neglected — 
Common  Method. 


or 


or 


.     (H5) 
(n6) 


(118) 


M,  =  —  /2Q419,  entering  table  with  I  —  k. 


(120) 


34  A    TREATISE   ON  AXCHES. 

V,=  !2n2Qk\i  -  k)*  ........     /(i  76)        (121) 

or 

(1210) 


or 


or 


(122) 
2k(i  —  k}\2  — 


or 

7S=/^J4»     reading  I  —  £  for  £  .........  (124^) 


or 

*t  =  ^i.»     reading  i  —  /^  for  /^  .........  (126^) 

^.  =  ^(3  -  12/fc  +  24^a  -  i6#)       ....      Xi83)        (127) 
or 


FORMULAS  FOR  PRACTICAL    USE.  35 

(d)  Horizontal  Loads,  with  Effect  of  Axial  Stress  included. 

if-™\-  +•  ^o) } '      X*87)      (128) 

where 


^)  .......     (,30) 

-  15  +  5o£  -60*"  +  24*')  =  4,-      •    (130 


-k- 


where  Hl  is  given  by  (128). 

(I33) 

#  =  ^.        •      (134) 

=z/3.      .    (I35) 


3  A    TREATISE   ON  ARCHES. 

k(l-k)  =  Ab  .........      (136) 

For  M,  read  (i  —  k)  for  k,  in  (132). 

^  =  1(^-^  +  20*)     .     (47)    .     (137) 

y*  =  jjt  .......          (50    •    ('38) 

Having  the  values  of  //,,  J/i,  Vlt  and  /,,  the  remaining 
outside  forces  are  readily  determined  in  the  manner  outlined 
for  vertical  loads  on  page  33. 

(<?)    Temperature. 


when  the  effect  of  the  axial  stress  is  included,  and 


when  the  axial  stress  is  neglected. 

•  D\ 
) 


.       5 

.    .     (141) 


when  the  effect  of  the  axial  stress  is  considered,  and 

^•=^°  =  ^      ....    /(I93)    •    •     (142) 
when  the  axial  stress  is  neglected. 

(I43) 


FORMULAS  FOR  PRACTICAL    USE.  37 

From/(i93)  and/(i9i), 

M*=M,=H?-f\       .........     (144) 


/095)    •    • 


(/)  Effect  of  a  Change  of  Al  in  the  Length  of  the  Span. 

.If  et°  be  replaced  b 
they  apply  to  this  case. 


If  et°  be  replaced  by  -j-  in  the  equations  for  temperature, 


(£")  Uniform  Loading. 
Let  w  be  the  uniform  load  per  unit  length  of  span.     Then 


);   .    .    .   /(2o6)    .    .    (147) 


.  W\k\-  i  +  3^  -  ^  +  ^  ;  .  /(2o8)  .    .  (148) 

^" 

....    /(207)  .     .  (149) 

;        ....     /(209)  .      .  (I50) 

^ 

^  -  Hj  -  ^  I  (2^r  _  a)a.  /(2io)  .  •  .  (151) 


3»  A    TREA  TISE   ON  ARCHES. 

In  some  of  the  equations  the  quantities  A  and  m  appear* 
For  A  an  average  value  of  £0  cos  0  may  be  taken,  and  for  m 
an  average  value  of  6/F. 

In  case  it  is  desired  to  take  advantage  of  the  effect  of  the 
axial  stress,  it  will  be  advisable  to  first  proportion  the  arch 
members  with  the  effect  of  the  axial  stress  neglected  ;  this 
makes  it  possible  to  take  nearly  correct  values  for  A  and  m  in 
a  second  computation  which  includes  the  effect  of  the  axial 
stress. 

*In  parabolic  arches  which  have  a  large  rise  in  comparison 
with  the  length  of  the  span  the  effect  of  the  axial  stress  upon 
the  values  of  Mt  and  H1  is  very  small, — in  fact  smaller  than 
errors  which  are  likely  to  be  made  in  graphical  solutions. 

In  flat  arches  the  axial  stress  may  be  of  considerable 
magnitude  (about  20  per  cent  for  //,  in  the  case  of  the  St. 
Louis  arch) ;  yet  this  is  greatly  reduced  when  the  stresses  are 
determined,  and  considering  that  a  safety  factor  of  from  four 
to  six  is  employed  in  proportioning  the  members,  the  degree 
of  danger  is  exceedingly  small  in  omitting  the  effect  entirely. 


B.  SYMMETRICAL  CIRCULAR  ARCHES. 


'(59)     • 


'See  Appendix  C 


FORMULAS  FOR  PRACTICAL    USE.  39 

9         /radius  of  gyrationV 
m=Flt*=\  ~7T~     "— /'        c(^  ('53) 


k'  =  R-f. C(6i)     . 

x  =  ^(sin  00  —  sin  0).    .     .         c(62)     .     (155) 
y  =  R(cos  0  —  cos  00).        .         ^(63)     .     (156) 


^  =  IT  -  *    +  (kf  +•??.  .        c(64)    .    (i 57) 


|/  —  x  „    .  ^ 

sm  0  =  — r> — ;   cos  0  =  — n — •      f(56)    .    (158) 


tan  0=^+j>/ ^(67)     .     (159) 

Since  the  general  method  of  treating  circular  arches  is  the 
same  as  for  parabolic  arches,  it  will  be  necessary  to  only  give 
the  equations. 

ARCHES   HAVING  TWO   HINGES,  ONE  AT  EACH  ABUTMENT. 

(a)   Vertical  Loads,  with  Effect  of  Axial  Stress  neglected — 
Common  Method. 

i^(sin"  00  —  sin*  a)  \ 

-f  cos  00(cos  <?-}-<*  sin  q— cos  00— 00  sin  00)  (. 

£(108)     (160) 
or 


,  =     P(i-k}     where     k  =  a/I.       ....     <r(m)     (161) 


4O  A    TREA  TISE   ON  ARCHES. 

y.  =  jja   ....     c(u$)     (163) 
or 

(163. 

(163*) 


(ft)    Vertical  Loads,  with  Effect  of  Axial  Stress  included. 
I  --  -  (sin"  00  —  sin*  a) 


,(117)  (164) 


j  -f-  g(00  -f  sin  00  cos  00) 

where 

i 

B  =  4s  —  2<&  cosS  0o  —  3  s"1  0o  cos  0o  H~  0o-  •     •     066) 

-f-  cos  00(cos  a  -\-  a  sin  a  —  cos  0,  —  0.  sin  00)  (167) 
or 

A  =4,4. (168) 

00  -f-  sin  00  cos  00  =  A^ (l^9) 

V,  =  2P(i  —  k).  .     .     .    c(ni)  (170) 

yt  =  -j£a.     .          .         .     c(nS)  (171) 


FORMULAS  FOR  PRACTICAL    USE.  4! 

(c)  Horizontal  Loads,  with  Effect  of  Axial  Stress  neglected. 

//  a  —  sin  of  cos  a 

H  =  --2Q  ]  I  +        —  2  cos  <?o(sin  a  —  a  cos  a) 

2         (          0.  —  3  sin  0g  cos  00  -|-  200  cos*  0~ 

C(\20)  (I72) 

a  —  sin  a  cos  a  =  >#, (*73) 

sin  a  —  a  cos  a  =  A  J18 (174) 

00  —  3  sin  00  cos  00 -f  200  cos' 00  =  JJg.  .     .  (175) 

r=-r.  =  i<2r.    •    •    •  (176) 


or 


sin  a  cos  a 
—  2  cos  00(sin  a—  a  cos  a) 

-     -  -  -  ' 


-T  -  :  -  3 

00  —  3  sin  00  cos 


20.  COS* 


or 


(177) 


(178) 


Horizontal  Loads,  including  Effect  of  Axial  Stress. 

f        00  —  3  sin  00  cos  0  n+  200  cos*  00  -f  «       ~\ 
r      j  —  sin  a  cos  a  —  2  cos  00(sin  a.  —  a  cos  a) 

{  +  w(0n  +  sin  00  cos  0o  +  a  -f  sin  a  cos  or)  i-        ^(125)    (180) 

I   2(00  —  3  sin  0n  cos  0n  -(-  200  cos*  00) 

I  -f  2w(00  +  sin  0o cos  0o)  J 


42  A    TREATISE   ON  ARCHES. 

or 


.     .    (181) 

(182) 


•  _:#,£ 

(f)   Temperature. 

efA  sm0, • 

•A.     <p  — 1—  200  cos   0 

—  3  sin  00  cos  00 

\  -j-  sin  00  cos  00) 


when  the  effect  of  the  axial  stress  is  included,  and 
,-,.        et°A  sin  0. 


when  the  effect  of  the  axial  stress  is  neglected. 

B  =  0o  +  200  cos2  0o  —  3  sin  00  cos  00  =  J18 .      .     (186) 
00  -j-  sin  00  cos  00  =  ^ia (l%7) 

Change  in  the  Length  of  the  Span. 
HI  =  _„,*    , ,~A.      . r^^       ^(130)     (188) 


when  the  effect  of  the  axial  stress  is  included,  and 

(I89) 


FORMULAS  FOR  PRACTICAL    USE.  43 

when  the  axial  stress  is  neglected,  Al  being  the  actual  change 
in  the  length  of  the  span. 

B  =  00  —  3  sin  00  cos  00 -f  200  cos' 0,  =  J18.       .     (190) 
00  +  sin  0e  cos  00  =  J, (191) 

SYMMETRICAL   CIRCULAR   ARCH   WITHOUT   HINGES. 

(a)    Vertical  Loads,  with  Effect  of  Axial  Stress  neglected — 
Common  Method. 

f  2  sin  00[cos  a  -\-  a  sin  a]  "1 

ff,  =  \ZP\    ~  %*  *&""  *'  +  *'  Si"  ^  \-     4-33)    (-92) 
I  0o*  +  0.  sin  00  cos  00  —  2  sin3  00  J 

cos  or  -f-  a  sin  a  =  //„.  .     .     (193) 

sin  00[2  cos  00  -f-  00  sin  0J  =  A^.  .     .     (194) 

0n3  +  0o  sin  0o  cos  00  —  2  sin8  0,  =  JM.  .     .     (195) 

J/i  =      '  •  (sin  0o  —  0o  cos  00) 

/ 

H — — r-. — -7 — T ^— :  \  sin  a00(cos  a  sin  00—  cos  00  sin  09— 00) 

^  200(sm  0o  cos  0o  — 0o)  < 

-j-  a00  sin  00 

-f- (sin  0o  cos  00  —  00)[cosa+asina— cos00— 00sin00]  \.    ^(134)    (196) 

sin  00  —  00  cos  0o  =  AA^.  .  .  .  (197) 

cos  a  -\-  a  sin  a  =  //„.  .  .  .  (198) 

—  cos  00  —  0,  sin  0o  =  An,  .  .  .  (199) 

—  (0.  —  sin  00  cos  00)  =  ^19.  .  .  .  (200) 

—  (0o'  —  0o  sin  00  cos  0.)  =  JM.  .  .  .  (201) 

(0ca  +  0o  sin  00  cos  00)  =  A^.  .  .  .  (202) 


44  A    TREATISE   ON  ARCHES. 

The  value  of  M,  can  be  found  from  (196)  by  assuming  a 
load  so  that  a  will  become  I  —  a. 


Mt  -4-  V,a 

- 


.(135)     (203) 


(50)    .    (204) 


H  -  ....... 

(50     •     (205) 


(52)    .    (206) 


Independent    equations    for  j0,  j,,   and  /,   are   given   in 
Chapter  IV. 


(b)  Horizontal  Loads,  with  Effect  of  Axial  Stress  neglected. 

^     /; 


{      r  00(sin  a  cos  a  —  a)        ^ 

-  +  ^Q\  ,    ,    +2sin00(Sinar-«cosa)  I 
1  2    1      r  2sin'00-00sin00cos0.  f 

I  -  0o'  J 

sin  or  cos  a  —  or  =  —  /?14  .....     (208) 

sin  a  —  or  cos  a  =  AA^  ......     (209) 

2  sin2  00  —  0.  sin  0,  cos  0e  —  0.'  =  —  JM.     .     (210) 

jy    r> 

i  =  —~  (sin  0.  —  0o  cos  00)    ..............     (211) 


i  __  ^Q^  _  \  (cos  a  -cos  00)(sin  <f>0  cos  00-  00+2  cos  a  sin  <£„)  ) 
«   2(sin  00  cos  00  -  00)  /  —  sin  00(sin»  00  —  sin*  a)  J 


) 

[  .    .     .     . 

) 

sin  0,  —  00  cos  0,  =  JJ,  ......     (213) 


sin  <f>,  —  0o  cos  0o  +  sin  a  —  a  cos  a    .   .     .     .    ^(143)    (212) 

) 


FORMULAS  FOR  PRACTICAL    USE.  45 

sin  0a  cos  00  —  0a  =  —  ftlt  .....     (214) 

sin  0a  —  0a  cos  0a  =  JJ19  .....     (215) 

sin  a  —  a  cos  a  —  AA^  .....     (216) 

Tie  magnitudes  of  //,  and  M,  can  be  found  from  (207)  and 
(212)  by  replacing  a  by  /  —  a,  etc. 

ri  =  l-(M,-Ml  +  2Qb).       c(  144)        (21  7) 
(£•)    Temperature,  with  Effect  of  Axial  Stress  neglected. 
i 


(2I8) 
0."  +  0o  sin  0o  cos  0a  —  2  sin8  00  =  JM.     .      (219) 

It  will  be  noticed  that  all  of  our  tables,  with  one  exception, 
have  been  computed  for  whole  degrees.  In  case  the  loads  do 
not  fall  at  even-degree  points,  it  will  be  found  advisable  to 
make  all  computations  for  .//,,  Mlt  Vlt  etc.,  for  the  even- 
degree  points,  and  then  obtain  the  values  corresponding  to  the 
true  positions  of  the  loads  by  reading  their  values  from  a  dia- 
gram constructed  from  the  calculations  thus  made. 

The  effect  of  the  axial  stress  has  been  omitted  here,  as  the 
equations  are  long  ;  these  are  given  complete  in  Chapter  IV. 

When  the  rise  of  the  circular  arch  is  not  greater  than  two 
tenths  the  span,  the  formulas  for  parabolic  arches  can  be 
applied  in  the  determination  of  the  external  forces  without 
sensible  error. 

Another  approximate  method  may  also  be  used  for  arches 
where  ///>  0.30,  viz.:  Substitute  a  parabolic  arch  of  the  same 
span  which  has  an  area  equal  to  the  area  of  the  given  circular 
arch  and  determine  the  external  forces,  and  then  apply  these 
forces  to  the  given  circular  arch. 

For  arches  approaching  a  semicircle  this  method  is  but  a 
few  per  cent  in  error. 


46  A    TREATISE   ON  ARCHES. 

C.  SUMMATION  FORMULAS  FOR  SYMMETRICAL  ARCHES  OF 
ANY  REGULAR  SHAPE  AND  ANY  CROSS-SECTION. 

ARCH   WITHOUT   HINGES. 
(a)    Vertical  Loads,  with  Effect  of  Axial  Stress  considered. 


^ 
^K'yAs       ^NXAX       o      ft,     ±yAs 

7     ^     h7   />         ^AS_  f  T 

(220) 


s       V  NxAx       p  ~B^  */  yAs 


where  f),  is  the  horizontal  thrust  due  to  two  equal  and  sym- 
metrically placed  loads. 
For  a  single  load 


where 


'  (22I) 


and 

AT*  A  T  a.   /f  •£• 

sin  0.    .     .     .     .     (224) 


FORMULAS  FOR  PRACTICAL    USE.  47. 

KxAs 


where 


?  =  ^^  cos  0  (approximately),       .     .     (228) 
1  x  o  •»** 

and  /^j  is  given  by  (221). 

rl  =  l(Mt-Mt+p(i-W).  .  .  .   (229) 

M 
y,  =jj  ............   (230) 

Having  //,,  Mlt  F,,  and  /,,  the  remaining  outside  forces 
can  be  found  by  the  method  explained  on  page  22. 

(b)   Vertical  Loads,  with  Effect  of  Axial  Stress  neglected. 

If  the  effect  of  the  axial  stress  is  to  be  neglected,  we  have 
merely  to  drop  the  terms  containing  Nx  and  Fx  in  (221)  and 
(225),  and  proceed  as  before. 

(c)  Horizontal  Loads,  with  Effect  of  Axial  Stress  included. 

H,  =    -.  +  <2).  .  .  ,  v.r 


A    TREATISE   ON  ARCHES. 


(232) 


where 


COS0.          «.»/-.«..  (235) 


cos  0     S. 


=.  (236) 


M9  for  a  load  situated  a  distance  a  from  the  origin  =  Ml 
for  a  load  situated  (/  —  a)  from  the  origin. 


(237) 
(238) 


Having  determined  //",,  J/,,  ^/a,  F",,  and  7,,  the  other  outer 
forces  are  readily  found,  as  explained  on  page  22. 


FORMULAS  FOR  PRACTICAL    USE. 


49 


(d)  Horizontal  Loads,  with  Effect  of  Axial  Stress  neglected. 

If  the  effect  of  the  axial  stress  is  to  be  neglected,  we  have 
merely  to  omit  all  the  terms  which  contain  F,  in  (236)  and  (232), 
and  proceed  as  before. 


(e)   Temperature. 
Eefl 


cos  ^ 


(239) 


yAs 


.      240) 


rt  =  o (241) 

If  the  effect  of  the  axial  stress  is  to  be  neglected,  omit  the 
terms  containing  Fx  in  (239)  and  (240). 


ARCH  WITH  A  HINGE  AT  EACH  SUPPORT. 
(a)  Vertical  Loads,  with  Effect  of  Axial  Stress  included. 


As 


\i     As          R    As  \ 

~" 


(242) 
(243) 


50  A    TREA  TISE   ON  ARCHES. 

If  the  axial  stress  is  neglected,  omit  the  terms  containing 
Fx  in  the  above  equations. 

(li)  Horizontal  Loads,  with  Effect  of  Axial  Stress  included. 


where 


(245) 


(c)   Temperature. 
Eet°l 


When  the  effect  of  the  axial  stress  is  neglected  the  terms 
containing  Fx  are  to  be  omitted. 

DEFLECTION   OF  ARCH. 

In  considering  the  deflection  we  will  assume  that  the  axial 
stress  is  neglected  and  that  no  change  takes  place  in  the  rela- 
tive positions  of  the  several  points  of  the  arch  other  than 
that  produced  by  the  loading. 

ARCH  WITHOUT  HINGES  OR  WITH  TWO  HINGES — 
(SYMMETRICAL  LOADING). 

JT+"°*-    •     e(&)    (247) 
\-et°y.-     .     .     g(6i)    (248) 


FORMULAS  FOR  PRACTICAL    USE.  5 1 

The  above  summation  formulas  are  sufficiently  accurate  for 
practical  purposes,  and  are  quite  simple  in  their  application. 
They  apply  to  any  regular  arch  figure,  such  as  circular,  par- 
abolic, oval,  elliptic,  gothic,  spandrel-braced,  etc.  They  are 
especially  useful  in  the  solution  of  the  spandrel-braced  arch, 
and  all  arches  which  have  variable  or  constant  moments  of 
inertia  not  following  the  laws  upon  which  the  formulas  of 
Chapters  III  and  IVare  based. 


CHAPTER  III. 

PARABOLIC   ARCHES,  WITH   THE   MOMENTS  OF  INERTIA 
VARYING  ACCORDING   TO  THE   RELATION 

A  =  £6  cos  0  =  a  constant. 


GENERAL   RELATIONS. 

IN  large  arches  it  is  convenient  often  to  arrange  the  sections 
so  that  their  moments  of  inertia  vary  according  to  the  relation 
A  =  EB  cos  0  =  a  constant.  This  assumption  enables  us  to 
deduce  quite  simple  formulas  for  the  determination  of  the 
reactions,  bending-moments,  etc. 

The  nomenclature  used  in  this  chapter  will  be  the  same 
as  heretofore  employed,  and  any  new  symbols  appearing  will 
be  found  clearly  represented  in  Fig.  24. 


FIG.  24. 
We  have  then 

A  —  E6  cos  0  =  a  constant. 


X59) 


Let 


e 


=  -~  =  (radius  of  gyration)*      .     .     .    /(6o) 

52 


PARABOLIC  ARCHES.  53 

and 

p  —  the  parameter  of  the  parabola. 

The  equation  of  the  parabolic  curve  referred  to  its  vertex  is 


or 


For  ^  =  0,^  =  0,  and  g  =  2//, X63) 

From/(62), 

y  ~  zp 

therefore 


and 

dy=*^dX    or    g  =  ^£  =  tan0.   .    .    /(65) 

From    </, 


If   J00  represents  any  change  in  00,  the  corresponding 
change  up  to  any  section  x  will  be  represented  by 


/(66) 


But  *.  =  —  L-;  hence 
dx      cos  0 


cos  0 


54  A    TREATISE   ON  ARCHES. 

A 

or,  since  E  cos  <£  =    -  (see  /(59)  )» 


J0  =  J0.  +  -L    f 

t/O 

From  (41), 

-a}-}-  2Q(y  -  b}.   (41) 


p-  _  x 

Substituting  (41)  in  p(6f),  remembering  that  y  —  -^—-  —  x 


(see  ^(64)  ),  we  have 

J0  =  J00  +  -L  ^ 
A  Jo 


-  -Li  |  PyT  V  -  «>&  I  +  j-i  {  Qf\y  -  b}dx  |  .      /(68) 

Performing  the  integrations  indicated,  factoring,  and  col- 
lecting, we  obtain 


.         - 
2A 


From  (#)  the  expression  for  J^r  from  o  up  to  the  section  X 
becomes 


dX.     .     .    /(7o) 


Substituting  the  value  of  ^0  obtained  above,  integrating 
and  reducing,  /(/o)  becomes  (see  Appendix  A) 


PARABOLIC  ARCHES.  55 


/(79) 


From  (^)  the  expression  for  Ay  from  o  up  to  the  section  x 
becomes 

Ay=  f  A<t>dx  +  ef  f*dy-±  f^y*     -  .  - 

^/O  t/O  i/O          * 


which  reduces  to  (see  Appendix  B) 


$6  A    TREATISE   ON  ARCHES. 

Equations  p(6g),  ^(79),  and  />(84)  are  general  expressions 
for  the  elastic  parabolic  arch,  symmetrical  or  non-symmetrical, 
when  EO  cos  0  is  constant.  They  can  be  employed  in  the 
determination  of  temperature  stresses  and  stresses  caused  by 
concentrated  loads  acting  in  any  direction  in  the  plane  of  the 
arch.  Those  terms  containing  the  factor  m  show  the  influence 
of  the  "  axial  stress." 

SYMMETRICAL  ARCHES—  GENERAL  FORMULAS. 

For  symmetrical  arches  these  equations  become  somewhat 
more  simple,  as 

g  =  \l    and     <p{  =  —  <pa 

Let  x  =  /,  and  assume  the  arch  to  be  symmetrical  ;  then 
<f>  =  ffif  =  —  00  and  y  =  c  =  o. 

Making  these  changes  in  /(6o,),  we  have 


- 


J*  -  H      -  2P(l  -  a)* 


From  (47), 

F/'  =  MJ-MJ+:SP(l-a)l  +  2Qbl.      .    /(86) 
Substituting  /(86)  in/>(85)  and  reducing, 

;  B  j*;  +  Z  I  jr,  +  jr.  -  H  !/+  1  M/  -  «)* 

2yi  (  / 


From  /(79)  we  obtain 


PARABOLIC  ARCHES. 


$7 


/(88) 


Equation  />(84)  reduces  to 
At  =  /J0.  +  ^+1^,-  ^,/+  ±$P\a(l- 


SYMMETRICAL  PARABOLIC  ARCH  WITH   A   HINGE  AT  EACH 
ABUTMENT. 


FIG.  25. 

In  Fig.  25  let  ABA'  represent   a  symmetrical   parabolic 
arch  having  a  hinge  at  A  and  A' ;  then  there  can  be  no  bend- 


$8  A    TREATISE   ON  ARCHES. 

ing-moments  at  these  points;  hence  Mt  and  Mt  =  o,  and  the 
resultants  Rv  and  Rt  will  pass  through  the  hinges. 

For  convenience,  the  effects  of  vertical  loads,  horizontal 
loads,  and  a  change  in  temperature,  with  and  without  omitting 
the  effect  of  Nx,  will  be  considered  independently. 

In  all  that  follows,  k  =     . 


(a)   Vertical  Loads,  with  the  Effect  of  Nx  omitted  — 
Common  Method. 

Assuming  that  /re-mains  constant,  J/=o,  and  by  remem- 
bering that  Mt  and  M,  =  o,  and  also  that  all  terms  containing 
Q  and  m  do  not  appear,  we  have  at  once  from  /(88),  by  solving 
for.fr,, 


or 


Values  of  k(i  —  2k*  +£')  are  given  in  Table  I. 

Since  all  loads  are  vertical,  the  horizontal  thrust  is  the 
same  throughout  the  arch. 

From  (39), 

HX  =  H,  ........     (39) 

making  x  =  /,  Hx  =  fft  ,  and  we  have  //,  —  //,  =  o,  or  Hl  and 
//,  are  equal  in  magnitude,  but  act  in  opposite  directions. 

From  /(91)  the  values  of  ff,  for  each  load  can  be  very 
quickly  found  with  the  aid  of  Table  I,  which  gives  the  values 
of  the  expression  k(i  —  2k*  -\-  k*)  for  values  of  k  from  o  to  i.oo. 

From  (47), 


PARABOLIC  ARCHES.  59 

or 


from  which  the  value  of  F,  for  each  vertical  load  is  readily 
obtained.  The  value  of  I  —  k  can  be  taken  directly  from 
Tables  I  or  V,  for  k  =  o  to  k  =  i.o. 

Having  H,  and   F,  ,  the  direction  of  Rs  for  any  particular 
load  is  found  from 


/(94) 


When  the  stresses  are  to  be  determined  by  graphics,  we 
need  only  to  use  ^(94)  and  determine  tan  /?,  for  each  load  ; 
then  since  Rt  for  each  load  must  pass  through  the  left  hinge, 
Rj  can  be  drawn  in  its  proper  position  at  once.  Since  RJ}  the 
load,  and  Rt  meet  in  a  point,  R9  must  pass  through  the  point 
of  intersection  of  R^  and  the  vertical  force  (load)  ;  it  must  also 
pass  through  the  right  hinge,  and  hence  its  direction  is  com- 
pletely determined.  The  values  of  //,  ,  //,,  F,  ,  and  V^  can 
now  be  found  by  simple  resolution  of  forces.  The  interme- 
diate stresses  can  be  found  by  Clerk  Maxwell's  method  of 
graphics  when  the  arch  is  trussed. 

To    facilitate   the    calculation   of    tan   /?,   the    values    of 

-  .         ,  —  T«  have  been  tabulated  in  Table  II. 
5  v1   i  *      &  ) 

From  (50)  we  have  for  each  load 


which  locates  the  point  of  intersection  of  Rt  and  R9,  making 
the  application  of  graphics  still  easier  than  the  method  using 


The  values  of  --      ,  _  „  are  given  in  Table  II  for  values 
of  k  from  o  to  i.oo. 


60  A    TREATISE   ON  ARCHES. 

From  (39),  (40),  and  (43)  we  obtain 

Tx=  (F,  —  2P)  cos  0  —  H,  sin 
From  (41), 

Mx  =  V,x  -  Hj  -  ZP(x  -a)  .....    /(97) 

By  means  of  p(gi),  /(93),  ^(96),  and  />(97)  tne  stresses  at 
any  point  of  the  arch  can  be  completely  determined  by  com- 
putation. 

Change  in  Shape  Due  to  the  Action  of  Vertical  Loads 
(Nx  omitted). 

From  /(89)> 

J0°  =  7  + 

The  term  -~  shows  the  effect  of  any  change  in  the  eleva- 

tions of  the  hinges.  This  does  not  mean  a  slight  difference  of 
level  in  the  hinges  before  the  arch  is  in  place,  but  any  change 
which  may  take  place  afterwards. 

In  construction  an  attempt  is  made  to  so  design  the  abut- 
ments, etc.,  that  Ac  will  be  zero. 

/>(98)  may  be  written  (Ac  assumed  zero) 


.    A99) 
From  /(69),  remembering  that  g  =  £/, 


From  /(79), 


PARABOLIC  ARCHES.  6  1 

From  ^(84) 

Ay  =  xJt.+fc  {  VlX-H^(2l-X)-±2P(x-aY  \  ,  /(iO2) 
in  which 

//,  =  |-£i/%(i-2£'  +  £')    .........      /(90 

and 

Vi  =  2P(i  -k)  ..............      /(93) 

(b}   Vertical  Loads,  Effect  of  the  Axial  Stress  included. 
From  X88)» 


H  = 
1  3  . 

«»- 


_  -  _ 

8//3+  30^0.1   3  2(/+2/) 

or 


in  which  £j  is  the  value  of  //,  given  by/(9l). 
Let/=  «/;  then 


Substituting  /(iO4)  i 


The  values  of  00  are  given  in  Table  XXV. 

*  HI  =  ^(i  —  e),     (approximately)    /(io6) 

where  ^  =  //",  as  given  by/(9i). 

*  See  Appendix  C. 


62  A    TREAl^ISE   ON  ARCHES. 

For  a  brief  discussion  of  the  effect  of  the  'axial  stress,  see 
Appendix  C. 

The  expression  for  Vt  is  not  affected  by  the  axial  stress ; 
hence 


k).  .    .......     /(93) 

For  any  load,  from  (50)  we  have 

y>  =  jja  =  fj-kl- 

From  (39), 


V.=  V,-2P,      ........        (40) 

Nx  =  Vx  sin  0  -\-  Hx  cos  0,      .     .     .     .        (42) 


(c)  Horizontal  Loads  (Nx  omitted}. 


FIG.  26 


In  Fig.  26  is  represented  a  single  horizontal  load  Q  acting 
from  the  right  to  the  left,  which  produces  a  horizontal  reaction 
H^  similar  in  character  to  that  produced  by  a  vertical  load 
acting  downward. 


PARABOLIC  ARCHES.  63 

From/(88),  for  any  number  of  horizontal  loads, 

p(no) 

.-    .    /(in) 

The  values  of  the  quantity  in  brackets, 


are  given  in  Table  III  for  values  of  k  from  o  to  i.oo. 
For  any  load  Q, 


From  (47), 


From  Fig.  26,  for  a  single  load, 

Vs0  =  HJ     or    *.  =  %•&  .....    /(1  15) 

From  (47),  for  a  single  load, 


Therefore 


64  A    TREA  T1SE   ON  ARCHES. 

or 


The  coefficient  of  /  is  given  in  Table  III. 
Having  the  value  of  xt  for  any  load  Q,  the  values  of  fflt  F. 
,  ,  etc.,  are  readily  determined  by  graphics. 
From  (39), 


From  (40), 

Vx—Vi />(ii8) 

From  (41), 

* 

From  (43), 

Tx  =  Vx  cos  0  —  Hx  sin  0 ; (43) 


(a  )  Change  of  Shape  Due  to  Horizontal  Loads 
(Nx  neglected). 

From  ^(89,  we  have 

o=  y  +  l^,-  i(2[i  -  2^(2  -  5*+  5^)  +  3*]}- 

The  coefficient  of  2Q  is  tabulated  in  Table  IV. 
From  /(/9), 


15(7--  30/^-5^]  I  .......     /(I22) 


PARABOLIC  ARCHES.  65 

From  X84)» 


in  which 


and 

V,  =  ±k(i  -k)n 


(e)  Horizontal  Loads,  Effect  of  the  Axial  Stress  included. 
From  X88), 


15 


5*vn 
J 


where 

i  15 


B      8//  +  30^/0; 


Here  we  see  that  the  effect  of  the  axial  stress  is  small.     If 
np(f>a  is  neglected  in  -„,  the  first 
of/>(i25)  at  once  reduces  to/>(m)- 


0  is  neglected  in  -^,  the  first  term  of  the  second  member 


66  A    TREATISE   ON  ARCHES. 

The  expression  J2  —  $k(i  —  k  —  2k*  -f-  ^/P)  +  8/£6}  may  be 

/         k  \ 

written    2\\  —  »  -[5(1  —  £  —  2&  -{-  4^)  -  8£4]J,    and   hence   its 

value  quickly  determined  from  Table  III. 

The  value  of  F,  is  not  affected  by  Nx  ;  hence 

V^  =  4k(i-k}n  ......    p(i  13) 

For  any  particular  load,  from  (58), 


(58) 


The  values  of  Hx,  Vx,  Nxt  and   Tx  are  given  by  (39),  (40), 
(42),  and  (43). 

Mx  =  V,x  - 


(/")  Change  of  Shape  due   to  the  Action  of  Horizontal  Loads, 
with  Effect  of  Axial  Stress  included. 

The  values  of  ^00,  Ax,  and  Ay  can  be  found  from  /(89), 
/>(79),  and  ^(84)  respectively. 

(^")   Temperature. 

A   change   in   temperature   is   equivalent   to    applying    a 
certain  horizontal  load  at  the  hinges  ;  or, 


if  the  axial  stress  is  neglected,  and 


if  the  effect  of  the  axial  stress  is  included. 


PARABOLIC  ARCHES.  6f 

A  rise  in  temperature  creates  a  reaction  H^  acting  from 
the  left  towards  the  right. 

The  values  of  Hx,  Vx,  Mx,  Nx,  and  Tx  can  be  found  from 
(39),  (40),  (41),  (42),  and  (43). 

(/z)  Change  of  Length  in  the  Span. 
From  X88)> 


neglecting  the  axial  stress  ;  or 

TT  6oA         f 

- 


if  the  axial  stress  is  included. 

If  the  span  is  shortened,  //",  acts  from  the  left  towards  the 
right. 

The  values  of  Hxy  VXJ  etc.,  can  be  found  from  (39), 
(40),  etc. 

(t)  Sinking  of  a  Support. 

In  case  one  of  the  supports  changes  its  elevation  after  the 
arch  is  in  place,  a  slight  change  in  the  stresses  may  result 
from  the  effect  of  the  change  in  the  length  of  the  span  ;  but 
any  change  likely  to  occur  may  be  neglected  in  the  calcula- 
tion of  stresses. 

(/)   Uniform  Loads. 

Thus  far  we  have  considered  only  concentrated  loads.  If 
the  load  is  uniformly  distributed  (horizontally), 

2P  =fwda  =  wlfdk, 
where  w  represents  the  load  per  unit  length  of  the  span. 


68 


A    TREATISE   ON  ARCHES. 


Let  Fig.  27  represent  an  arch  having  a  partial  uniform 
load;  then,  from/(90» 


From  /(97), 


= T[  I  ^(2  - k)  -  £^a(5  - 


From/(96), 


,     where    ^  =      .    /(i3S) 


PARABOLIC  ARCHES.  69 

The  above  equations  enable  us  to  determine  all  the  stresses 
in  the  arch  when  the  axial  stress  is  neglected. 

(k)  Uniform  Load  Over  All. 

In  case  the  load  is  distributed  horizontally  and  uniformly 
over  the  entire  span,  then  k"  =  o  and  k'  =  i  or  x/l,  and  we 
have, 


If  n  =  o,  //i  =  o. 

From/(i33), 

V  —  — 

From /( 1 34), 

*  ~    2  24* 

If  ft  =  o,  then  jj/  =  o,  and 


the    expression    for   the   bending  -  moment   in   the    ordinary 
straight  girder. 
From /(i  3  5), 

T-  =  —  cos  0  —  -z—ivl  sin  0  —  wx  cos  0.        /(i39) 
2  8« 

If  «  =  o,  0  becomes  zero,  since  our  radius  is  now  infinity; 
hence 

^       wl 

Tx  =  —  -  wx, 

the  expression  for  shear  in  the  ordinary  straight  girder. 


A    TREATISE   ON  ARCHES. 


SYMMETRICAL   PARABOLIC  ARCH   WITHOUT   HINGES. 
LocuslLine         3 


FlG.    28. 

In  this  case  we  have  several  conditions  which  must  be  sat- 
isfied. As  in  the  case  of  the  arch  with  two  hinges,  we  shall 
consider  the  various  loadings,  etc.,  independently. 

(a)    Vertical  Loads  (Nx  neglected). 
From/(87),  by  transposition, 

-  a)', 


and  from  /(88)» 


Now  if  there  are  no  hinges  the  ends  of  the  arch  must  be  fixed 
in  direction,  and  00  =  <&  cannot  change  under  any  condition  of 
loading. 

If  the  length  of  the  span  be  assumed  unchanged  and  the 
effect  of  temperature  omitted,  we  have,  by  combining  ^(140) 
and/(i4i)  and  reducing, 


PARABOLIC  ARCHES, 


or 

H>  =  l£sP#(i-Kr  .......    X'43) 

The  values  of  k?(i  —  Kf  are  given  in  Table  XI  for  values  of 
k  from  o  to  i.  oo  inclusive. 

From  ^(89),  assuming  that  Ac  and  J00  are  zero, 


M,  +  \M%  =  HJ  -          P(l  -  a)(2l  -  a)a.    .  /(i44) 

Combining  /(i4i)  and  />(i44),  assuming  Al  and  et°lto  be  zero, 
we  obtain  by  reduction 


or 


Substituting  the  value  of  Hl  from  /(  143),  we  have 


The  values  of  k\i  —  ^)($  —  5/£)  are  given  in  Table  VI  for 
values  of  k  from  o  to  I.OO  inclusive. 

Substituting  (i  —  k}  for  k  in/(J47)»  we  have 

Mv=- 

The  values  of  k(i  —  k)\e>k  —  2)  are  given  in  Table  VI   for 
values  of  k  from  o  to  i.oo  inclusive,  reading  (i  —  k)  for  k. 


72  A    TREATISE   ON  ARCHES. 

From  (47), 


Substituting  /(!47)  and  ^(148)  in  /(I49),  we  obtain  by 
reduction 


The  values  of  (i  -  k?(i  +  2/£)  are  given  in  Table  VII  for 
values  of  k  from  o  to  i.oo  inclusive. 

The  above  equations  completely  determine  all  of  the  ex- 
ternal forces.  The  stresses  at  any  section  of  the  arch  can  be 
determined  from  equations  (39)  to  (43)  inclusive,  remembering 
that  the  terms  containing  Q  disappear,  as  we  are  not  consider- 
ing the  horizontal  components  or  loads. 

The  values  of  //,,  Vlt  and  Rt  can  be  found  graphically  after 
the  ordinates  y^  ,  j0  ,  and  ^  are  determined. 

From  (50),  (51),  and  (52), 

M,  +  V,a  Mt  Mt 

y.  =  .  '  y>  =  jr>   and  ^== 


Substituting  the  values  of  fft,   V^,  etc.,  given  above,  we 
have 


(positive  upward)    .     . 


PARABOLIC  ARCHES.  73 

which  completely  determines  the  position  of  the  equilibrium 
polygon  for  any  vertical  load. 

When  j,  orj/2  become  very  large  it  is  more  convenient  to 
use  the  abscissas  x^  and  x^. 

From  (54)  and  (55), 


xl  is  negative  when  measured  towards  the  left. 


2(3  -  2*) 

x^  is  negative  ivhen  measured  towards  the  right. 
The  coefficients  in/(i52)  and  /( 153)  are  tabulated  in  Table 
VIII,  and  those  in/(i54)  and/(i55)  in  Table  IX. 

(b)  Change  of  Shape  due  to  the  Action  of  Vertical  Loads 
(Nx  neglected). 

From  /(89), 


where  M^,M^,  and  Hl   are  to  be  found  from  /(i48),  /(i47), 
and/>(i43)  respectively. 

The  values  of  k(2  —  *$k  +  P)  =  2k  —  3^  -f-  ^'  are  given  in 
Tab)e  X  for  values  of  k  from  o  to  i.oo  inclusive. 


in  which  Ml  ,  V,,  and  H,  are  to  be  found  from  /(i48),/(i5o), 
and  /(  143)  respectively. 


74  A    TREA  TISE   ON  ARCHES. 

From  /(/p), 


where  Yl/,  ,  F,  ,  and  //",  are  to  be  found  from  ^(148), 
and/>(i43)  respectively. 
From  ^(84), 


-  a)3   , 


where  y)/,  ,  F",  ,  and  //,  are  to  be  found  from  /(i48),  /(i5o), 
and  /(  143)  respectively. 

These  four  equations  completely  determine  the  change  of 
shape  due  to  the  action  of  any  vertical  load. 

(c)    Vertical  Loads,  with  Effect  of  Axial  Stress  included. 


a).      .    ......    /(i6o) 

From  />(88)' 

>  =  Htf  -    ,2P(l  -  a)(r  +  al-  a^a 


PARABOLIC  ARCHES. 


Subtracting  p(i  60)  from  /(l60  and  solving  for  Hlt  we  have, 
by  reduction, 


where 

c= 


The  values  of  &(i  —  £)*  and  k(i  —  k)  are  given  in  Tables 
XI  and  V. 

We  have,  approximately, 


where  ^  =  //,  in/(i63). 
From/(89), 


-- 

The  first  member  of  /(l63)  may  be  written 


or.f 


multiplying  /»(i6i)  by  D,  we  have 


*  Appendix  C. 


A    TREA  TISE   ON  ARCHES. 


M.D  +  MJ)  =  #;—  /  -  ~ 


Eliminating  M,  from  X1^)  and 


in  which  Hl  is  to  be  found  from  ^(162)  and  Z>  from  (m). 
For  values  of  k(\  -  2/fea  +  /^3),  see  Table  I  ;  2k  —  $&  +  /P,  see 
Table  X  ;  and  for  /&(i  -  K)  see  Table  V. 

It  is  to  be  noticed  that  D  and  the  coefficients  containing 
m  are  constant  for  any  particular  arch,  hence  by  the  aid  of  the 
tables,  /(i68)  can  be  evaluated  very  rapidly. 

The  value  of  M1  can  be  obtained  from  ^(164)  by  taking 
everywhere  (i  —  k)  for  k,  or  by  first  computing  the  value  of 
Mt  from/(i64)  and  substituting  in/(i6i). 

The  value  of  Vl  can  be  found  from 


The  stresses  at  any  point  of  the  arch  can  now  be  deter- 
mined from  (39)  to  (43)  inclusive,  remembering  that  all  terms 
containing  Q  disappear. 

The  values  of  y<>,  y^  and  y^  can  be  found  from  (50),  (51),  and 
(52),  if  graphics  is  employed  in  determining  the  intermediate 
moments  and  shears. 


PARABOLIC  ARCHES.  77 

(d}  Change  of  Shape  due  to  Vertical  Loads,  including  Effect  of 
Axial  Stress. 

A(f>,  Ax,  and  Ay  can  be  found  from  /(69)>  P(79\  and  /(84> 
respectively,  remembering  that  all  terms  containing  Q  dis- 
appear, and  that  Ac,  Al,  and  J00  are  zero. 

(e)  Horizontal  Loads  (Nx  neglected}. 
From  X87)» 


V  -  6at*  -  40?) 

From  /(88), 

+  M,  =  Hf  -          Qa(l  -  a) 


s 
Equating  /(l65)  and/(l66)>  and  solving  for  Hlt  we  have 


or 


in  which  the  quantity  in  [  ]  is  tabulated  in  Table  XII. 

Eliminating  M,  from  /(88)  and  A89)'  and  solving  for  M, 
we  obtain 


A    TREA  TISE   ON  ARCHES. 


/' 

or 
and 


where  the  expression  in  j  \  is  tabulated  in  Table  XIII. 
Substituting  (i  —  k)  for  k  in/(i7i)  and  changing  sign, 


where  the  expression  in  {  }  is  tabulated  in  Table  XIII,  reading 
(I  -  K)  for  & 
From  (47), 

.     .    .    /(I73) 


Substituting  /(i  74),  X1  72),  and  /(i  70 

*)'^    ....    /(i75) 


or 


The  values  of  (/&  —  £')'  can  be  found  from  Table  XI. 

The  above  equations  completely  determine  the  external 


PARABOLIC  ARCHES.  79 

forces.     The  stresses  at  any  point  of  the  arch  can  be  found 
with  the  aid  of  (39)  to  (43)  inclusive. 

The  method  of  graphics  may  be  employed  after  we  have 
found  the  values  of  yl ,  y^ ,  xl ,  x^ ,  and  x^  in  determining  the 
external  forces. 


From  (51)  and  (52), 

y^  —  _| \_i\      ~~  _i_      A~    f^p    i        /.'i^    P{1^} 

and 


_y,  and  y^  are  always  measured  upward, 
The  coefficients  of  /in/(i78)  and  ^(179)  are  tabulated  in 
Table  XIV. 

From  (54)  and  (55), 


^r,  w  always  measured  towards  the  left  and  x^  towards  the 
right. 

The  coefficients  of  /  in  /(  1  80)  and  /(  1  8  1  )  are  given  in  Table 
XV. 

From  Fig.  29, 


or    ;r  = 


8O 


A    TREA  TISE    ON  ARCHES. 


Substituting   the    values   of  Hl  ,   F,  ,   and   b  from  /(i68), 
/(i76),  and  />(  1  74),  we  have 


).  .     .     .    /(i  83) 


•„  w  always  measured  from  left  to  right. 


FIG.  29. 

The  values  of  3  —  \2k-\-  24^'  —  i6/^3  are  given    in  Table 
XVI. 

'/)  Change  of  Shape  due  to  Horizontal  Loads  (Nx  neglected). 
From  ^(69),  since  J00  =  o, 


From 


PARABOLIC  ARCHES. 


8  1 


From  ^(84), 


where  Mt,  V,  ,  and  //,  are  given  by/(i7i),/(i76),  and/(i68) 
respectively. 

(g)  Horizontal  Loads,  with  Effect  of  Axial  Stress  included. 

From  ^(87)  and  /(88),  in  a  manner  similar  to  that  employed 
for  vertical  loads,  we  have 

\-  1  5  +  &k 


where 


The  values  of 


are  given  in  Table  XII. 


82  A    TREATISE   ON  ARCHES. 

From/(88)  and/(89), 


from  which  the  value  of  J/,  can  be  found. 
(i  -  4/fc  +  io/&a  -  io/63  +  3^)  =  i  -  2k(2  - 
and  the  values  of  this  expression  are  given  in  Table  IV. 


and  the  values  of  this  expression  are  tabulated  in  Table  III. 


and  /fj  is  given  by/»(i87). 

The  values  of  k(i  —  k)  are  given  in  Table  V. 

It  is  to  be  noticed  that  D  and  the  coefficients  containing  m 
are  constant  for  any  particular  arch  ;  hence  by  the  aid  of  the 
tables,  />(i8g)  can  be  readily  evaluated. 


PARABOLIC  ARCHES.  83 

The  value  of  Ml  can  be  found  from/(i89)  by  taking  every- 
where (i  —  k)  for  k. 

The  value  of  Vl  is  found  from 


/(I49) 


The  stresses  at  any  point  of  the  arch  can  now  be  determined 
from  (39)  to  (43)  inclusive,  remembering  that  all  terms  con- 
taining 2P  disappear. 

The  values  of  y9,ylt  etc.,  can  be  found  from  (50),  (51),  and 
(52),  if  graphics  is  employed  in  determining  the  intermediate 
stresses 

(A)  Change  of  Shape  due  to  Horizontal  Loads,  including  Effect  of 
Axial  Stress. 

A<(>,  Ax,  and  Ay  can  be  found  from,  ^(69),  ^(79)  and  ^(84) 
respectively,  remembering  that  all  terms  containing  2P  disap- 
pear, and  that  Ac,  A  I,  and  A<f>^  =  o. 

(i)  Temperature. 
From/(8/)  and/^88), 


where  the  term  containing  m  shows  the  effect  of  the  axial  stress. 
If  this  be  omitted, 

A*A 

Hl  =  ^--et0    (axial  stress  neglected).  .    .    ^(191) 
47 

Substituting /(icp)  in/(88)  and/(89),  and  solving  for  Mt, 
letting  />=I+ 


84  A    TREATISE   ON  ARCHES. 


If  the  axial  stress  be  neglected,  D  becomes  unity,  and  the 
terms  containing  m  disappear. 

Mt  =  ~-et°  =  MI     (axial  stress  neglected)    .    /093) 
or 


and 


The  intermediate   stresses,  etc.,   can  be   found  from  the 
general  equations  (39)  to  (43)  inclusive. 

(/)  Effect  of  a  Change  Al  in  the  Lrngth  of  the  Span. 

An  inspection  of  the  general  equations  /(87),  /(88),  and 
^(89)  shows  that  --  —  follows  the  same  law  as  -j-  et°.     Hence 


and 

HI  =  —  ^TTi^A  (neglecting  axial  stress) 


PARABOLIC  ARCHES.  8$ 

also 


or.  if  the  axial  stress  be  neglected, 

M,  =  -  l^Al  =  Mt  =  \HJ,  .....    /(200) 
where 


r 

The  intermediate  stresses  can  be  found  from  (39)  to  (43) 

inclusive. 

(£)  Effect  of  any  Change  in  00,  <&,  and  the  Relative  Positions 
of  the  Supports  in  Elevation. 

From/(8;)  and/(88), 


or,  if  the  axial  stress  be  neglected, 

^  =  -^(^0.-^0o) X202) 

From/(88)  and/(89),  by  substituting /(2Oi), 


86  A    TREATISE   ON  ARCHES 

If  the  effect  of  the  axial  stress  be  neglected, 


and 


(/)   Uniform  Loads. 

Using  the  same  nomenclature  as  employed  in  discussing 
this  case  for  the  two-hinged  arch,  we  have,  from/(i43), 


From  p(  147), 

k> 


From  /(  1  48), 


From  /(  1  50), 

yf 

r.^jJfc-aJP-T,*)  ......   X209) 

>i" 
From  (41), 


PARABOLIC  ARCHES.  8/ 


(m)    Uniform  Load  Over  AIL 


Here  k"  —  o  and  kf  =  I  or  x/l. 
From/(2o6), 


From/>(207), 

~Sn 
M,  —  o.  . 

From/(2o8), 

M  —  o.  .    .    .    .    .    . 

From  ^(209), 

V.-™' 

From/(2io), 

"wl        ivl         wx* 


/(2I2) 


CHAPTER  IV, 


CIRCULAR  ARCHES  HAVING  ~  =  A  CONSTANT. 


GENERAL  RELATIONS. 


Let 


FIG.  30. 


_  -pa 

A  —  -7j—  =  a  constant  ;  .......    ^(59) 


k'  =  R-f.       ...........  <r(6i) 

Then  from  the  equation  of  the  circle 

x  =  g  —  R  sin  0  =  R(s'm  <P0  —  sin  0);  .    .  c(62) 

y  •=  R  cos  0  —  k'  =  fi(cos  0  —  cos  00);     .  ^(63) 


CIRCULAR  ARCHES.  89 


a-  —  x 
sin  0  =  ^--  ,     cos  0  =  —      i;       ....    <r(66) 


From  (X  ),  for  any  point  x 


/*M 
•^ts, 


snce 


hence 


But  from  (41), 

^  =  M,  +  F^  -H,y-  2P(x  -  a}  +  i0(^  -  *).      (41) 
Therefore 


The  several  integrals  have  the  following  forms 


rF^0  =  /*  F.^(sin  00  —  sin 
*/*0 

-  00)  +  V,y 


9O 


A    TREATISE   ON  AKCHES. 

cos  </>  d(j)  —*HtR  cos 


—  R  sin  0  - 


-  cos  a)d<t> 


Substituting  the  above  values  in  ^(71)  and  ^(68), 


*T2 


From  («), 


After  substituting  the  value  of  A$  from  <r(77)  in  (^),  the 
following   integrals   will   aid  in  the   reduction    of  the    term 


jf  V.  - 


-*;   ....   478) 

a  -  sin  0)sin 


CIRCULAR  ARCHES.  QI 

Therefore 

'—  ^*(00— 0)  I ;  £(79) 


r$P(y 

t/O 


sin 

P(g  -  a)<t>R  sin  0^0 

-  a)\(a  -  <ft(y  +  k')  -  (x-  a)\;  . 
-  b}dy  =  2P<JW.  . 


I    2(21?  (sin*  4>—  sin  a  sin  0—  cosor^sin  0+a  cosasin  0X0 


r*  -AT- 

Integration  of    j    -^—dx 
i/o    ^f* 

r^*=  r 

J,    EF'          Jo 


/4  A*  #  >4  T 

and  = 


.     But  R  =  —  -j=\  hence  we  have,   after  substituting 


—  •  j    .  —  -j= 

the  value  of  NM  as  given  by  (42), 


92  A    TREATISE   ON  ARCHES 


But  from  (39)  and  (40), 

//,  =  #,-  i<2 


and 


hence  the  second  member  of  ^(82)  becomes* 


where 


.  -  0) 
cos1 


x)dx  =2PR*  sin  0  cos 

)1  .......    ,(86) 


CIRCULAR  ARCHES, 

Therefore  (a)  becomes 


93 


-  *\ 


4-  k'x  +  gy-  ^  (P.  —  0)] 


+  (0  -« 


x-gy  + 


From  (3), 


.  .(87) 


-dy.     .    ,(88) 


The  following  integrals  are  employed  in  reducing  ,(88): 
=  -  {^(0o  -  0)  -  ^(0o  -  0)  -  *!: 


94 


TREATISE   ON  ARCHES. 


(g-  d)(y  -  b)\; 
kf)dx 


or 


+  b)\(g-  X}(a  -  0)_(,  _ 


?  sin  0  cos 


in  which  the  following  integrals  occur  after  substituting  the 
values  of  Vx  and  Hx  from  (39)  and  (40): 


sin  0  cos 


CIRCULAR  ARCHES. 


95 


sin 


-  «)  +  #(*  -  «)  -  ^O  -  J)  +  xy  -  *£}; 


/   2QR*  si 


sn      cos     i      = 


Using  the  above  integrals,  ^(88)  become 

Ay  =  efy  +  xA<p 


k'x  - 

-a}(a-  0)  }  - 
a)\b  +  k'-\  +  (  y  - 


-  x)(a  -  0) 


Equations  c(77),  ^(87),  and  <r(99)  are  perfectly  general  for 
circular  arches,  and  can  be  applied  for  any  loading,  either  ver- 
tical or  horizontal.  The  equations  also  enable  us  to  consider 


A    TREA  TISE   ON  ARCHES. 


arches  which  are  not  symmetrical.     The  equations  for  sym- 
metrical arches  are  considerably  more  simple. 

SYMMETRICAL  CIRCULAR  ARCHES. 

For  this  case  we  have  g  =  £/,  and  for  x  =  /,  y  =  c  =  O,  and 

0  =  0J  =   —  0.« 

<r(77)  now  becomes,  remembering  that 


y«""»      -""i 
i  °*  A 


-  2aa  -  /0  - 


.    .    <T(IOI) 


From 


-a-ba-  k\a  +  00)}        _j      ^IQ2) 
From  ^(87)  we  have,  when  x  =  /, 


+  2P\a(l-a-2k'a}-k'(l(t>t-la-2b)\ 
2*"  a  2^  -  / 


-  a) 


CIRCULAR  ARCHES. 


97 


from  which 


/  -  2k '0. 


+  2P\a(l-a-2k'a)-k'(l<t>,-l<x-2b}\ 


-f 


From  ^(99), 


20)  +  (I  -  2a)(a 


where 


d  = 


A    TREA  TISE   ON  ARCHES. 


SYMMETRICAL  CIRCULAR   ARCHES   WITH   A   HINGE  AT   EACH 
SUPPORT. 


'     FIG.  31. 

In  Fig.  31  let  ABA'  represent  a  symmetrical  circular  arch 
having  a  hinge  at  A  and  A'  \  then  there  can  be  no  bending- 
moments  at  these  points ;  hence  Ml  and  Mt  are  zero,  and  the 
resultants  R,  and  Rt  will  pass  through  the  hinges. 

As  in  the  case  of  parabolic  arches,  we  will  consider  each 
class  of  loading,  etc.,  separately. 


(a)    Vertical  Loads,  ivith  the  Effect  of  Nx  neglected. 

Assuming  that  /  remains  constant,  Al  =  o;  and  by  remem- 
bering that  M!  and  Mt  are  zero,  and  also  that  all  terms  con- 
taining Q  and  m  do  not  appear,  we  have,  from  ^(87), 


_      /X/  -a-  2k'  a)  -  k'(l<f>0  -  la-2b)\  . 
1  ~  ~^70,  +  2*V0-3£V 

or,  since  a  =  R(sin  <p,  —  sin  or), 
b  =  7v?(cos  a  —  cos  00),     k'  =  R  cos  00,     and     /  =  2R  sin  <£c. 


CIRCULAR  ARCHES.  99 

c(io6)  becomes 

(sin*  0a—  sin*  a)  —  2a  cos  00(sin  00—  sin  a) 

—  2  cos  00[(00—  a)  sin  004~  cos  00—  cos  a] 

20,  cos"  00  —  3  sin  00  cos  00  -j-  0f 

r  i-(smS  0o  —  sm* 

_  4D^+COS  0." 


The  values  of  -g-  are  given  in  Table  XVII 

Since   all   loads  are  vertical,  Hl  and  fft  will  be  equal  in 
magnitude. 
From  (47), 


or 


where  k  —  a/I. 

^r(uo)  can  also  be  written 

^    sin^+^a 
2  sm  0. 

Having  determined  the  values  of  /T,  and  Vlt  the  stresses 
in  the  arch  can  be  found  by  graphics  or  by  means  of  equations 
(39)  to  (48). 

From  (50),  for  a  single  load, 


100  A    TREATISE   ON  ARCHES. 

or 


By  means  of  4114)  the  curve  DE  in  Fig.  31  can  be  located 
and  the  stresses  in  the  arch  found  graphically. 

The  values  of  A/B  can  be  found  from  Table  XVII,  and  of 
k(i-k}  from  Table  V. 

Change  of  Shape  due  to  the  Action  of  Vertical  Loads 
(Nx  neglected). 

The  values  of  A<f),  Ax,  and  Ay  can  be  determined  from 
<r(77),  ^(87),  and  <:(99)  Dv  remembering  that  all  terms  contain- 
ing Q  and  m  disappear,  that  Ml  and  Mt  are  zero,  that  g  =  ^/, 
and  that  the  values  of  Hl  and  Vl  are  to  be  found  from  ^(109) 
andr(in). 

(b)   Vertical  Loads,  Effect  of  the  Axial  Stress  included* 
From  <r(87), 

~  *  ~2"  ~  k>(1^  ~lU~  2^} 

.      . 


H  = 
1       1  -m\2Pa(l-a)\  J 


-k'l) 

or 


2  B  -j-  2w(00  -f-  sin  00  cos  00) 
or 

I r(sin*  00  —  sin*  «)    ' 


I  4-     (0»  +  sin  00  cos  00) 


See  Appendix  C. 


CIRCULAR  ARCHES.  IOI 

in  which  1}  is  to  be  found  from  ^(109)  and 

A  =  \  (sin2  0o  —  sin2«) 

-j-  cos  00(cos  a  +  a  sin  a  —  cos  <f>0  —  00  sin  0^ 

B  =  2  00  cos2  00—3  sin  00  cos  00  +  00. 

Since  the  value  of  B  is  constant  for  any  particular  arch, 
the  values  of  A  can  be  very  easily  found  from  Table  XVII  by 
multiplying  the  tabular  quantities  by  B. 

The  denominator  of  c(\i^)  is  constant  for  any  particular 
arch,  and  hence  the  value  of  //,  can  be  found  with  but  little 
labor. 

The  value  of  F,  can  be  found  from  c(m). 

From  (50), 


where  V,  and  H^  are  to  be  found  from  c(m)  and  c(\i?)  re- 
spectively. 

For  practical  purposes  it  will  be  sufficient  to  compute  but 
a  few  values  of  y^  and  then  draw  the  curve  D£,  Fig.  31,  by 
means  of  a  curved  ruler. 

The  change  in  shape  of  the  arch  can  be  found  by  means  of 
487),  and  c(99). 

(c}  Horizontal  Loads  (Nx  neglected}. 
From  ^(87), 

t      t    (^  +  2^)(0o  +  «)  +  ^(2«-/)    ) 
,^-\-^Q\  -  &'(!  -a)  +  2bk'ot  __  I 

(  2R\<p9-  3  sin  00  cos  00+2  0,  cos1  0Q)  ) 


a  —  sin  a  cos  a  \ 

^       -  2  cos  00(sin  a  —  a  cos  a)     (  .  <r(i2o) 

00  —  3  sin  00  cos  00  -j-  2  00  cos8  0,  ) 


IO2  A    TREATISE   OAr  ARCHES. 

The  values  of  00  —  3  sin  00  cos  00  -j-  200  cos1  00  are  given 
in  Table  XVIII;  of  a  —  sin  a.  cos  a.  and  sin  a  —  a  cos  «,  in 
Table  XIX. 

From  (47), 


V,=   -V,  ..........     '.       f(  1  22) 

Having  the  values  of  Vl  and  Hl  ,  the  stresses  can  be  found 
graphically  or  by  equations  (39)  to  (48). 

As  in  case  of  the  parabolic  arch,  we  can  locate  the  locus  of 
the  points  of  intersection  of  R,  and  ^  by  means  of  the  formula 


Substituting  the  values  of  //,  and  V^  from  ^(120)  and  c(i2i) 
for  a  single  load,  we  have 

!a  —  sin  a  cos  a  \ 

I  +        -gcos&fstnflf-gcos*)     [  ,  ,(123) 
00  —  3  sin  00  cos  00  -f-  2  00  cos  00  ) 

which  is  easily  evaluated  by  means  of  Tables  XVIII  and  XIX. 
The  change  in  shape  can  be  determined  from  ^(77),  ^(87),  and 

Horizontal  Loads,  including  Effect  of  Axial  Stress. 
From 


0_  -  J 

/a 


or 

0o  —  3  sin  00  cos  0o  ~h  2^*o  cos!l  ^o  ~h  a      1 
—  sin  a  cos  a  —  2  cos  00(sin  a  —  a  cos  a)  L 
+^l<?!)o+sin00COS004-flr+sinacosflrl     J       ,       . 
00  -  3  sin  00  cos  00  +  200  cos'  00) 
+  2w(00  -f  sin  00  cos  00) 


CIRCULAR  ARCHES.  103 

which  can  be  quickly  evaluated  by  means  of  Tables  XVIII 
and  XIX. 


and 


in  which  //",  and  F,  are  to  be  found  from  4I25)  and  c(i2&). 

The  change  in  shape  can  be  determined   from  ^(77), 
and  <99). 

(^)   Temperature. 
From  <:(87)  or  ^(103), 

«•  _  <?A  _  sin  «A.  __ 
l~    R    00  +  200  cosa^0  -  3  sin  00  cos  09 

H-  w(00  +  sin  00  cos  00) 

or,  when  the  effect  of  the  axial  stress  is  neglected, 

,,       efA  sin  00 
- 


<r(i28)  and  c(  1  29)  are  quickly  evaluated  by  the  aid  of  Tables 
XVIII  and  XIX. 

(f)  Change  in  Length  of  Span. 
From  ^(87), 


m  (0a  +  sin  00  cos  00) 
or,  if  the  effect  of  the  axial  stress  is  neglected, 


These  equations  are  readily  evaluated  by  the  aid  of  Tables 
XVIII  and  XIX. 


104  A    TREATISE   ON  ARCHES. 

(g}  Sinking  of  a  Support. 

In  case  one  of  the  supports  changes  its  elevation  after  the 
arch  is  in  place  a  slight  change  in  the  stresses  may  result  from 
the  change  in  span,  but  this  usually  will  be  too  small  to  be  of 
any  practical  importance, 

SYMMETRICAL   CIRCULAR   ARCH   WITHOUT   HINGES. 

• 
(a)   Vertical  Loads  (Nx  neglected}. 

Equating  ^(102)  and  <r(IO4)  and  solving  for  //,,  we  have 


_  -  1(1  -  2a)(00  -a)-  t  .       . 

2l{k'<i>>  +  d<t>   -1} 

which  reduces  to 

;      ,f  2  sin  00  [cos  a  -f-  «  sin  a] 

Hl  =  ^P-(  —  sin  00  [2  cos  0fl  -f-  00  sin  00]  —  0e  sin*  or  [-,£(133) 
00*  H~  0o  sin  0o  cos  0o  ~ 


which  is  easily  evaluated  by  the  aid  of  Tables  XX,  XXI,  and 
XXII. 

Substituting  <r(iO2)  in  ^(105),  and  then  solving  for  Mlt  we 
have 

M,  =  ™.  (sin  0o  -  0o  cos  00) 


"1  —  ..  /  .  —  ^—    —  ;  r  -  -T-  1  sin  or00(cos  a  sin  00  —  cos  00  sin  00  —  00) 
200(sm  0o  cos  0o  —  00)  i 

-j-a0csin0o+(sin00cos0e  —  00)[cosa+  asinar  —  cos00—  00  sin00] 

By  the  aid  of  Tables  XIX,  XXIII,  and  XXIV  ^134)  can 
be  quickly  evaluated. 

The  value  of  Mt  can  be  found  from  ^(134)  by  assuming  the 


CIRCULAR   ARCHES.  1 05 

load  applied  at  a  point  on  the  arch,  so  that  (a)  in  ^(134)  will 
become  (/  —  a). 
From  (47), 


-^,     .     .     4,35) 

where  k  =  -.' 

Having  determined  the  values  of  ffltMlt  and  F.,  the  stresses 
can  be  found  graphically,  or  by  equations  (39)  to  (41). 

The  ordinates  fixing  the  locations  of  the  resultants  Rt  and 
R^  for  any  particular  load  can  be  found  from  the  following 
equations. 

From  (50),  (51),  and  (52), 


and 


(50) 


From  (51)  and  (52)  we  obtain,  by  substituting  the  values 
of  Ml  and  Mt,  and  remembering  that  Hl  =  .//,  in  magnitude, 


sin00cos00-00(+sin  00(cos  a  sin  a 
and 

y*  +  Ji  =  3-(sin  ^o  —  0.  cos  00) 


t0. 

V  3  ' 


sm  a  —  cos     0  —    C  sn  00^ 
From  ^(136)  and 


106  A    TREATISE   ON  AXCHES. 


y\  =  :r(sin  00  —  0o  cos  0o) 


C002  +  0o  sin  0o  cos  0o  —  I 

-.«  *>  {[..„  *.cos  *  -  W[_  ™7.±™L%.]  U 
-)-0o  sin  00(cos  a  sin  «+a)—  sin  a(00  cos  00  sin  00+0os)  [•  _ 

0o(sin  0o  cos  0o  —  0o)  1  2  sin  0o[cos  a  -f-  a  sin  a] 

-  sin  0U[2  cos  00  +  00  sin  00]  —  00  sin2  a  J 

By  the  aid  of  Tables  XX,  XXII,  XXIII,  and  XXIV  ^(138) 
can  be  readily  evaluated. 

Evidently  yt  can  be  obtained  from  (138)  by  making  (a) 
equal  (/  —  a). 

From  (50), 

.   sin  0.—  sin  a,  \  \    P  sin2  ^o  —  sm2  a  ™ 


77^^  change  in  shape  can  be  found  from  ^(77),  ^(87),  and 

^(99)- 

(^)  Horizontal  Loads  (Nx  neglected). 

From  ^(102)  and  ((104), 


_//-  4_  T4- 

2  r  2  sin2  00-  00  sin  00  cos  00—  002)  4  ' 

which  can  be  easily  evaluated  by  means  of  Tables  XIX  and 
XX. 

Substituting  c(iO2)  in  ^(105),  and  eliminating  M^  between 
^(102)  and  ^(105),  we  have 


CIRCULAR  ARCHES. 

or 

IT  n 

Ml  —  — l—  {sin  00  —  0,  cos  <pt\ 

00 


, —  -&-- j(cos  01  —  cos  00)(sin  00  cos  00  —  00 

'  2(sin  00cos  00— 00)  |-f-  2  cos  asin  <P0) —  sin  00(sin2  00 — sin2") 
i 

:=1—  {sin  0g  —  00  cos  00  -f-  sin  a  —  a  cos  «},    .     .     .     ^(143) 

which  can  be  evaluated  by  the  aid  of  Tables  XIX  and  XXIII. 
The  magnitudes  of  //,  and  Mt  can  be  found  from  ^(141)  and 
^(143)  by  making  (a)  equal  (/  —  a). 
From  (47), 

j  / 

/ 

Having  the  values  of  Hl ,  H^ ,  F,,  Vtt  Mlt  and  M, ,  the 
reactions  R^  and  Rt  are  completely  determined,  and  the  stresses 
can  be  found  graphically  or  by  equations  (39)  to  (41). 

The  change  in  shape  can  be  found  from  ^(77),  ^(87),  and 
499)- 

(^r)  Effect  of  a  Change  in  Temperature,  Length  of  Span,  the 
Angle  00 ,  etc. 

From  ^(102)  and  ^(104), 

1     "/7      2k'<j>\A<t>i—'A<f>} 

^d  _  /)  ^(HS) 

or 


_  -  -          0  -        0  , 

1  .  sin  00  cos  00  -  2  sin3  00)       '    ^  4 


where  the  value  of  the  parenthesis  in  the  denominator  can  be 
obtained  from  Table  XX. 


108  A    TREA  TISE   OX  ARCHES. 

From  c(iO2)  and  ^(105), 


A 


rr  r> 


A  sin  00 


cos  0o  _  0f) 


f  -{-^(200sm00-|-sin00cos00— 00) 

\  4"  ft(2<Po  si"  ^o  —  si"  ^0  cos  *A)  +  ^o)  j  . 

(  sin  00  ° 

which  can  be  evaluated  by  the  aid  of  Table  XIX. 
From  (47), 


.(149) 


The  stresses  can  be  now  found  from  (39)  to  (41). 
The  change  in  shape  can  be  found  from  ^(77),  ^(87),  and 
<99). 

(d)  Effect  of  the  Axial  Stress. 

In  order  to  economize  space,  the  expressions  for  H^  and  M^ 
will  be  given  which  are  perfectly  general,  applying  to  cases  of 
vertical  loads,  horizontal  loads,  change  in  temperature,  etc. 

From  4102)  and  1(104), 


-  1(1- 


—  2m<t>92Pa(l  -  a) 


0  +  or)  -  k'l  —  %bl+  a(b  +  k')\ 


CIRCULAR  ARCHES.  109 

The  terms  containing  m  show  the  effect  of  the  axia  Istress. 
From  c(iO2)  and  ^(105), 


+  2^V00  +  (kf  —  d}(i  +  m\2b  —  2aa  —  /0,  +  /a) 
-  m\$t(l  -  2a)(b  +  d)-  2a</>0R*]  } 


_  -LsQ\l  -a-  tec-  k'(a+  0.)} 

20, 


The  terms  containing  m  show  the  effect  of  the  axial  stress. 

By  the  application  of  c(\$Q),  <i5i),  and  (47)  the  stresses  in 
any  symmetrical  circular  arch  without  hinges  can  be  completely 
determined  by  the  ordinary  methods  of  graphics. 


CHAPTER  V. 

SYMMETRICAL    ARCHES    HAVING    A  VARIABLE  MOMENT 
OF   INERTIA. 

THE  treatment  of  symmetrical  arches  can  be  considerably 
simplified  by  the  methods  we  are  about  to  introduce.  The 
following  equations  for//,  and  Ml  can  be  applied  to  any  sym- 
metrical arch  when  the  axis  is  a  curve  which  can  be  expressed 
by  a  linear  equation.  We  will  first  consider  the  case  where 
the  arch  has  no  hinges. 


FIG.  32. 

SYMMETRICAL  ARCH   WITHOUT   HINGES. 
Value  of  Mr—  From  (d),  (a),  and  (b), 


°  /   dx  -  ±  I   ^dx. 

Jo  ^-Jo     *» 

>    f  dy-±   f^dy. 

i/t)  »/  0  * 


SYMMETRICAL  ARCHES. 


Assume  a  single  load  placed  at  any  point  upon  the  arch; 
then,  since  the  arch  is  fixed  at  the  ends  and  symmetrical,  J0; 
=  J00,  Al—  O,  and  Ac  =  o.  If  x  =  /,  ^(59),  ^(60),  and  g(6\) 
become,  neglecting  temperature  for  the  present, 


J-  /  ~^s= 


and 

^__j.     C1M^ 

Now,  from  (41), 


where 


Substituting  ^-(65)  in  g(&2)  and  ^-(64),  we  have 


and 


rids  .    .  clxds,  ciKdS 
,    r+rtt  -e-+    ~r  =  o 

i/O       *  i/O         *          Jo        V* 

Fxch  r**3         ClKXds        C1N 

*  I  -^+r*  I  —+  I  -Q  --  /  jr 

t/O          *  i/O  *  t/0  *  »/J    •r* 


From  (47), 


where 


112 


A    TREATISE   ON  ARCHES. 


Substituting  the  value  of  Vl  in  ^(67)  and  ^(68),  and  elimi- 
nating My,  we  have 

(  rK*ds__  fljv   i  clxds    riKds  rix*ds 

\   I       0  IF  y\   I    ~Q    '     I     01    ~rO~ 

Ti/r    .  v*/0  x •/  0         x        JI/Q  «y  0          x  »y  0          * 


in  which 


sin  0  +  ^  cos  0, 
-  -  />,         ^  >  a.     From  (40) 
,-Q.        x>a.     From  (39) 


(42) 


^(73) 


Then  in  g(ji)  we  have  two  unknown  quantities,//",  and  Vx. 
But  Vx  occurs  in  Nx  only,  which  contains  the  effect  of  the  axial 
stress  ;  hence  for  the  common  method  of  arch  treatment  we 
can  neglect  the  term  containing  Nx.  A  method  will  be  given, 
however,  which  will  enable  us  to  very  nearly  obtain  the  actual 
effect  of  the  axial  stress. 

Value  of  //,.  —  The  vr.lue  of  fft  can  be  found  as  follows: 


FIG.  33. 

Assume  the  arch  free  to  slide  longitudinally  upon  the  sup- 
ports, and  that  two  equal  and  symmetrical  loads  are  applied ; 
also  assume  that  there  are  equal  and  symmetrical  moments 
Hz  applied  at  the  supports;  then  /J0,  =  ^0,  the  same  as  if 


SYMMETRICAL   ARCHES.  113 

the  arch  were  fixed  at  the  ends,  since  our  loading  is  symmetri- 
cal.    From  £(62)  we  have 


But  from  (41), 

where 

K'  =  F^  -  2P(x  -a)  +  2Q(y  -  b),    .    .    ^76) 

Hj  being  zero,  since  the  arch  is  free  to  slide  upon  the  sup- 
ports. 

Substituting  £-(75)  in 


or 


ClMds        C1M+K'  Clds         C1K' 

I  -7T  =    I    '-*-£ — ds=Ml        &-+        -$-<**  = 
JQ          *         J  i/o  i/o       * 

/K' 
-x-ds 
u 
* 


The  change  in  length  of  the  span  due  to  the  action  of  our 
loading  can  be  found  by  the  aid  of  ^63).  Let  A'l  be  the 
change  in  the  length  of  the  span  ;  then 


Substituting  the  value  cf  Ml  from  ^(78), 
ClK'ds 

jv=  ii-t  A*-2  r^^- J 


A    TREA  TISE   ON  ARCHES 


where 


K'  =  Vtx  - 


-  a] 


=  F,  sin  0  +  /f,  cos  0 ; 


(42) 
(40) 

(39) 


All  of  which  are  known  quantities ;  hence  the  value  of  A' I 
can  be  accurately  determined  from  ^(80)  for  any  symmetrical 
loading. 


FIG.  34. 

Now  suppose  the  arch  unloaded  and  free  to  slide  as  before, 
and  let  two  equal  and  symmetrical  moments  Q'z  be  applied  at 
the  supports ;  then  J0;  =  ^00,  and  we  have  from  ^(62) 


/I 
Q* 


o. 


But 
hence 


SYMMETRICAL   ARCHES.  115 

or 


rlya 
J0     0 


yds 

g(»3) 


The   corresponding  change  in  the  length  of  the  span  is 
given  by  £(63),  or 


where   Mx  =  Q'(z  +  y)  and  Nx  =  +  Hx  cos  </>  =  +  Q  cos  0; 
hence 


Substituting  ^83)  in  £(85), 

r 


Let  ^j  be  the  horizontal  thrust  at  the  support  necessary  to 
cause  a  change  in  the  length  of  the  span  of  ^7;  then  we  have 

^/:J7::>:^  =  ^~ 

Therefore 


V, 


I   ds 

Jo  J* 


A    TREATISE   ON  AKCHES. 


where 


K'  =  V,x  -  2P(x  —  a 


x=a. 


-  b)  ; 


For  two  equal 
and  symmet- 
rical loads. 


(a)  Vertical  Loads  only. 
If  the  loads  are  vertical, 


=  Px  - 


For  two  equal 

and  symmetrical 

vertical  loads, 


-  a) 

-  a)  ; 


Vx=  V,  -2P=  P  -  2P-, 
Nx=  V,  sin  0—  2P  sin  0 
=  P  sin  0  —  2P  sin  0. 


Since  our  loads  are  equal  and  symmetrically  placed,  and 
K'  is  the  moment  at  any  point  x,  considering  the  arch  as  an 
unconfined  girder,  the  value  of  K'  due  to  one  load  will  have  a 
corresponding  equal  value  due  to  the  other  load.  Then,  since 

there  are  symmetrical  values  of  -^-,  the  value  of   /  K'y-r^  for 

one  load  must  be  equal  to  that  for  the  other  load. 
Therefore  for  a  single  vertical  load  we  have 


K'  =  P(i  -  k}x  -  [/>(*  -  a)  when  x  >  a~\ 


SYMMETRICAL  ARCHES. 


117 


and 

Nx  =  P(i  —  k)  sin  (f>  —  [P  sin  0  when  x  >  a]. 

For  x  =  o  to  ^  =  a, 

K'  =  />(i  —  %     and     Nx  —  P(i  -  k)  sin  0.  .    ^(88) 
For  x  —  tf  to  ;tr  =  /, 

AT'  =  Pk(l-x)    and     ^  =  -  /£  sin  0  ;    .     .    ^(89) 

and  we  have 


From  ^(90),  the  horizontal  thrust  due  to  any  vertical  load 
can  be  found  when  the  relation  between  x  and  y  is  known. 
The  equation  applies  equally  well  to  the  parabolic,  circular,  or 
elliptic  arch. 


Horizontal  Loads  only. 


Here  we  have 


K'  = 


and 


For  two  equal  and  sym- 
metrical horizontal 
loads,  x  >  a. 


=  -  2Q  cos  0. 


118 


A    TREA  TISE   ON  ARCHES. 


For  x  —  o  to  x  =  a,, 

K'  —  o  and  Nx  =  o ; g(9l) 

for  x  =  at  to  x  =  ^a , 

JST'  =  0(j  —  ^)  and  Nx  =  —  Q  cos  (j> ;  .  ^(92) 
for  x  —  a^  to  -r  =  /, 

7T'  =  o    and     Nx  =  o ^(93) 

Let  /f,  =  the  thrust  due  to  the  load  on  the  left ; 

Ht  =  the  thrust  due  to  the  right  load. 
Then 


but  H^+l 

hence      2/7,  =  £ 
or 


Q 


Therefore 

r 

2  + 


or 


-I- 


/:i 


i  — 


z? 

/»<£* 
—  cos  0 

/>".  (y  -  b}ds 

—  «*        &*       r'yfo 


SYMMETRICAL   ARCHES.  119 

is  general,  and  can  be  applied  to  parabolic,  circular, 
and  elliptic  arches  with  equal  facility. 

(c)  Moments,  Vertical  Loads  only. 
In  £-(71),  for  a  single  vertical  load, 

Kl=-Hj-P(x-a]        x  =  a, 
and 

Nx  —  F,  sin  <{>  —  P  sin  0  -f  H^  cos  0,        #;;«, 

where  //,  can  be  found  from  ^(90). 

There  remains  then  only  the  term  F,  sin  0,  which  is  as  yet 
unknown.  In  case  there  are  equal  and  symmetrical  loads  V, 
becomes  known,  as  it  is  equal  to  one  half  the  total  loading. 

If,  however,  the  loading  is  not  symmetrical,  the  values  of 
Ml  and  Mt  can  be  computed  with  the  term  F,  sin  0  neglected, 
and  the  corresponding  value  of  F,  found  from  (47),  and  then  a 
second  calculation  made  and  this  value  introduced.  Generally 
the  value  of  the  expression  containing  F,  is  very  small,  and  is 
omitted  entirely  by  nearly  all  American  authors. 

Neglecting  the  term  containing  F,  ,  for  x  =  o  to  x  —  a, 


and 

Nx  —  H^  cos  0  (approximately);  ..... 
for  x  =  a  to  x  —  /, 

K=-Hj-P(X-a}  ........ 

and 

Nx  =  Hl  cos  0  —  Psin  0  (approximately).  . 
Therefore  £-(71)  becomes 


/  A 

"  U 


120 


A    TREATISE   ON  ARCHE3. 


where  the  value  of  H^  is  to  be  found  from  ^90).        J 

The  value  of  Mt  can  be  found  from  g(ioi)  by  replacing  a 
by  (/-«). 

(d)  Moments,  Horizontal  Loads  only. 
In  the  case  of  horizontal  loads  only, 


Nx  =  Vx  sin  0  +  Hl  cos  0  —  -2<2  cos  0. 
Then  for  x  =  o  to  x  =  a, 


and 

Nx  =  /^,  cos  0  (approximately); 

for  x  —  a  to  x  —  /, 


and 

jV*  =  /^,  cos  0  —  <2  cos  0  (approximately). 

Therefore  ^(71)  becomes 


where  //;  is  to  be  found  from  ^(95). 


SYMMETRICAL   ARCHES.  121 

The  value  of  Mt  can  be  found  from  ^(106)  by  replacing  a 
by  (I  -a). 

(e)  Effect  of  a  Change  in  Temperature. 
Assuming  that  the  span  does  not  change  in  length, 


=  et°l  =  o; 
or,  if  the  arch  is  free  to  slide  upon  the  supports, 


Let  Ht  be  the  horizontal  thrust  necessary  to  cause  a  change 
in  the  length  of  the  span  of  A'l-,  then,  referring  to  ^(86), 


or 

Eefl 


From  ^(83), 

/i  yds 

r-  -  -  ~^~ 
t/ 

But     M,  =  H& 


122 


hence 


M1  =  -. 


A    TREATISE   ON  ARCHES. 


/  /«xyy 

r^co-0       ^ 

/ds 

«/   vx  H 

^c°      /I 

.   Q* 

SYMMETRICAL  ARCH   WITH   A   HINGE  AT   EACH   SUPPORT. 

In  this  case  we  have  no  morrtents  at  the  points  of  support. 
Assume  that  the  arch  is  free  to  slide  upon  the  supports 
then,  from  ,^(63), 


FIG.  35. 
Let  Q  be  any  horizontal  load  at  the  hinges  ;  then 


and 

Nx  =  Hx  cos  0  =  Q  cos 

Then  ^(113)  becomes 


S I 'MME TRICA L   AR CHES. 


123 


Now  suppose   the   horizontal   loads  Q'  removed  and  two 
equal  and  symmetrical  vertical  loads  applied  to  the  arch  ;  then 


E  t/o 

x 

Mx  =  V,x  —  2P(x  —  a) 

and         F,  =  i/'Ci  -  £); 
hence 


Nx  —  Vx  sin  0  +  //^  cos  (p  =  Vx  sin 
But  Fx  =  F,  -  2P;  therefore 

Nx  =  2P(i  —  k)  sin  0  —  2P  sin  0. 


Then  for  x  =  o  to  x  =  a, 


Mx  = 


and 


—       sn  0; 


124  A    TREATISE   ON  ARCHES. 

for  x  =  a1  to  x  =  at, 


and 

N,  =  2P(i  —  k)  sin  0  —  Ps'm  0;    .     .     .    ^(123) 
for  x  =  #,  to  x  =  /, 


and 

A^  =  -2/\i  —  /&)  sin  <f>  —  2Psin  0.  .    .     . 

Evidently  the  change  in  the  length  of  the  span  due  to  the 
left  load  will  equal  that  due  to  the  right  load  ;  hence  we  have, 
for  x  =  o  to  x  =  #, 


and 


for  x  =  a  to  ;r  =  /, 

J/.  =  />(!  -  k)x  -  P(x  -  a)  .     .     .     .    ^(128) 
and 

Nx  —  P(i  —  k]  sin  0  —  Psin  0.     .      .    ^(129) 

Therefore  ^"(117)  becomes 

™  <t>dx 


P     rt(x--jfyds      P    flsin  <t>dx 

~E  J    -  0  ---  ^EJ    ~~F  -  '       '     ' 

*<  »/ja  v»  •£•  */  a  r  x 


SYMMETRICAL  ARCHES. 


125 


Let   f^,  represent  the  horizontal  thrust  necessary  to  cause 
a  change  in  the  length  of  the  span  of  A"  I]  then 


and  we  have,  since  for  vertical  loads  the  horizontal  thrust  is 
constant  and  //,  =  ^,  , 


dx 

—  COS  0 


From  g(i$i]  the  horizontal  thrust  due  to  any  vertical  load 
can  be  found  with  comparatively  little  labor. 


i 

rT — 1   I01 


FIG.  37. 

For  two  equal  and  symmetrical  horizontal  loads  we  have 
from  (41).  assuming  the  arch  free  to  slide, 


Mx  =  V.x  - 

but  Vl  =  o, 

hence 

Mx  =  2Q(y  -  b\ 


126  A    TREATISE   ON  ARCHES. 

Nx  —  Vx  sin  0  -\-  Hx  cos  0  =  Hx  cos  0  ; 
but 

//,  =  //;-  i& 

hence 

Nx  =  -  2Q  cos  0.  ......        .     . 

Then  for     ;r  =  o  to  ;r  =  #,  ,  and  for  x  =  at  to  x  =  /, 

^,  =  0   ............    ^-(134) 

and 


for  x  =  rt,  to     JT  =  #,, 


and 


Hence  the  corresponding  change  in  the  length  of  the  span  is 


Then   since 


SYMMETRICAL   ARCHES. 

For  a  single  load,  //,  =  ££),  +  %Q.     Hence 


-—COS  0 


an  equation  quite  simple  in  its  application. 

*  If  the  moment  of  inertia  is  assumed  to  vary  according  to 
the  laws  assumed  by  most  writers  upon  the  theory  of  arches, 
their  equations  can  be  very  easily  obtained  from  our  general 
forms. 

For  example,  let  the  horizontal  thrust  //,  for  a  single  vertical 
load  placed  upon  a  parabolic  arch  having  no  hinges  be  required. 
Assuming  0  cos  (f>  =  A  =  a  constant,  and  that  the  terms  contain- 
ing the  effect  of  the  axial  stress  are  neglected,  and  remembering 
that  ds  cos  0  =  dx,  we  have,  from  £"(90), 


+  (I  _ 


-  x]dx 


f 

I/O 


dx 


/  yd* 

Jo 


dx 
where,  using  the  nomenclature  employed  in  Chapter  III, 


-  x)ydx  = 


*  For  several  examples  illustrating  the  application  of  these  general  formulas 
to  special  cases,  see  Appendix  E. 


128  A    TREATISE   ON  ARCHES 


/***  =  /, 

t/O 


and  we  have 


\fl\k  -  2k'  +  f)  -  \fl\k  -  ff)  „ 

ffl  =  ~- 


or 


which  is  the  same  as  obtained  by  the  method   employed   in 
Chapter  III.     (See  equation XJ43)»  Page  71-) 

In  a  similar  manner  any  of  the  equations  usually  employed 
can  be  quickly  deduced  from  our  general  formulas,  which  have 
the  advantage  of  being  general  to  the  extent  that  they  can  be 
employed  for  any  arch  when  the  relation  between  x  and  y  can 
be  represented  by  a  linear  equation. 

SUMMATION  FORMULAS. 

In  many  cases  it  is  preferable  to  replace  the  sign  of  integra- 
tion by  that  of  summation.  This  is  particularly  true  in  arches 
where  the  moments  of  inertia  do  not  change  according  to  some 
law  which  permits  of  readily  reducing  the  above  equations  to 
fit  the  paTficurar  case.  As  examples  of  such  structures  may 
be  mentioned  the  Douro  Arch  and  the  Washington  Bridge. 

The  summation  formulas  are  as  follows : 


SYMMETRICAL  ARCHES.  12$ 

(A)   ARCH   WITHOUT  HINGES. 
Vertical  Load  only. 


.  .   A* 


Horizontal  Loaa  only. 


*  "^7"^r  ~  vf 


130 


A    TREATISE   ON  ARCHES. 


0, 


0      P*  0 


Temperature. 
Eefl 


ARCH  WITH  TWO   HINGES  (ONE  AT  EACH   SUPPORT). 
Vertical  Load  only. 

^  —&-  sin  <p 


(x-  d]yAs 


z  Ax  ^ 

+  2-^-  sin  0  \ 

a     rx 


Horizontal  Load  only. 


—  ^  -pr  COS  0 


Temperature. 
Eefl 


SYMMETRICAL   ARCHES. 


13* 


ARCH  WITH  ONE  HINGE  AT  THE  CROWN. 

This  type  of  arch  is  seldom,  if  ever,  employed  by  American 
engineers.  French  and  German  engineers  sometimes  consider 
masonry  arches  having  lead*  or  iron  hinges  at  the  skew-backs 
and  the  crown  as  one-hinge  arches  for  moving  loads. 

For  this  case  we  will  neglect  the  effect  of  the  axial  stress  as 
being  of  little  importance  in  cases  where  this  form  of  arch 
would  be  employed. 

Vertical  Loads. 

Value  ofH^. — Let  two  equal  and  symmetrically  placed  loads 
be  applied  to  the  arch ;  then  ^00  =  A<f>i. 


From  ^(63)  we  have 


where 


ClMxyds 
=   \  —^-  =  o,      . 

i/U 


When   x  =  —  ,  Mx  =  o,    since  there    can   be    no   bending 


moment  at  the  hinge  ;  hence 


.    £(151) 


See  pages  229  and  230. 


132  A    TREATISE   ON  ARCHES. 

But  since  our  loads  are  equal  and  symmetrically  pkced, 
F,  =  P,  and 


Substituting  this  value  in  £"(150)  and  then  the  value  of  Mx 
in  ^(149),  we  have 


or 

'yds 


C^_ 

J,  ^  y. 

T'^  {'yds 

,  -».    '      'J.    ». 


-   r-      /yg   .Tr- 

J.   «.  "JJ,  »* 

where  at  =  /  —  at. 

Value  of  F,  and  F,  /<?r  Vertical  Loads. 
From  ^(61)  we  have 


which  becomes,  for  x  =  — , 


SYMMETRICAL   ARCHES.  133 


Let  two  equal  and  symmetrically  placed  loads  be  applied 
to  the  arch  ;  then  F,  =  P,  and  for  our  two  loads  we  have, 
from  (41), 


-a}.     .    .     .    g(it>7) 
But 


Hence 


Then  if  ^,/be  the  vertical  displacement  of  the  crown  due 
to  the  action  of  these  two  loads, 


//*    ,  /J/a 

f- 


For  a  single  vertical  load, 

-V^  +  HJ+'^P^-^  .......    g(i6i) 

Therefore, 


If  /J,/be  the  vertical  displacement  of  the  crown  due  to  a 
single  load,  we  have 


134  A    TREATISE   <9A'  ARCHES. 


.pT(,-af*-v,L 

•SO  x  I/O  „    „ 

Since  the  vertical  deflection  of  the  crown  due  to  one  of 
two  equal  and  symmetrical  loads  must  be  one  half  that  due  to 
both  loads,  J.,/=  2^,/.  Equating  these  two  values  and 
solving  for  Vl ,  we  obtain 

///'                 /•//«  ~//» 

*-£-P        (*-af*-P        ^1 
v  =_ ff*  J* °*  Jo       #*       ,;.*., 

1      2  /      /''/1  /W/* 

_   I    xds  _   /    art 

2. A  -T    Jr.  T 


tw  equation  is  to  be  employed  for  all  loads  on  the  left  of  the 
crown. 


These  equations  enable  us  to  find  the  values  of  V^  and  Vt 
for  all  loads. 

Values  of  M^  and  Mt  for  Vertical  Loads. 

From  (41),  making  x  =  -  and  solving  for  J/,,  we  have,  for 
a  single  load,  ' 


in  which  the  values  of  V,  and  77,  are  given  by  ^164)  and  ,^(154). 

This  equation  gives  the  values  of  Ml  for  any  load  on  the 
left  of  the  crown. 

From  (49), 

M  =  M       V    -Pl-a 


SYMMETRICAL   ARCHES. 

HORIZONTAL   LOADS. 
Value  of  H^  for  a  Single  Horizontal  Load. 


'35 


FIG.  39. 

Let  two  equal  and  symmetrically  placed  horizontal  loads 
act  upon  the  arch ;  then  Vl  =  o  and  (41)  becomes 


If  x  =  -,  then  Mx  =  o;  hence 


and 

^  =  9,/-  §J-Q(f- 
From  £-(63), 


Substituting  the  value  of  Mx  and  solving  for  I},,  we  have, 

&     /*•/„    ^yds 


f*lys         riy'ds 

J0  ~^~JQ~e^ 


I36  A    TREA  TISE   ON  ARCHES. 


Value  of  F,  for  a  Single  Horizontal  Load. 

This  case  will  be  treated  in  a  manner  similar  to  that  em 
ployed  for  vertical  loads.     From  £"(156), 


From  (41),  for  two  equal  and  symmetrical  loads, 


But 


hence 


The  vertical  displacement  due  to  two  equal  and  symmetrical 
loads  is 


I  (  fixds  fi 

*f=Mf       Tx~^ 

(          Jo  i/o 


xyds 


Q(f-  *) 


For  a  single  load, 


-Q(f-b}+Q(y-b}-V?-2 


SYMMETRICAL   ARCHES.  137 

B"t  *f>=m  +  Q>\       .....     ^(180) 

hence 


and 


Equating   the   two  values  of  AJ  and  solving  for  F,  ,  we 
obtain 


-  Qf        +  Q     *i 


x*ds  _  /_    rixds 
°*    "  Vo      '• 


which  reduces  to 


/'^rVj  _   /_    ^7^' 
*m     "~   *  I         9* 
J o 


which  holds  good  for  #//  /^^r  <?«  M^  left  of  the  crown. 


138  A    TREATISE   ON  ARCHES. 

Values  of  M1  and  Mt  for  a  Single  Horizontal  Load. 
From  (41), 

M,  =  -  V±  +  HJ-  Q(f~  fy      •     •    £(i85) 
and  from  (49), 

M^M.  +  VJ-Qb  .......    £<i86) 

Temperature. 

Assuming  that  Al  =  o,  and  that  the  hinge  at  the  crown 
remains  midway  between  the  supports,  we  have, 


From  (41), 

Mx  =  M,-H,y,  .....    £(188) 

but  for  x  =.  —  ,  Mx  =  o,  and  hence 


Therefore 


Substituting  this  value  of  Mx  in  ^"(187)  and  solving  for  H^  , 
Eet°l 


y«/o   e~ 

The  above  equations  are  perfectly  general,  and  in  their 
integral  form  can  be  applied  to  any  symmetrical  arch  which 
has  a  regular  curve  for  an  axis.  In  the  summation  form  the 
equations  apply  in  the  case  of  any  symmetrical  arch. 


SYMMETRICAL   ARCHES.  139 

If  the  axis  is  parabolic  in  form  and  Ed  cos  0  =  A  =  a  con- 
stant, our  equations  become  quite  simple. 

SYMMETRICAL     PARABOLIC     ARCH    WITH     A     HINGE    AT    THE 
CROWN  AND   £6  COS  0  =  A  CONSTANT. 

(a)  Single  Vertical  Load. 

/»/  /»/  * 

P  I    (x  —  a,}ydx  —    I    2P(x  —  d]ydx 

TT  *          *A) «£ 

/2     = — ; ; • 

2 


where 

%P(x  -  d)ydx  = 

Substituting  the  value  of  y  and  integrating, 


From  ^-(164), 

(I-  a)  f^xdx  -    C\x  -  d]xdx  -    fix* 


,  -- 

i-  fixd*  -  r 

2  t/o  t/o 


^(197) 


140  A    TREATISE   ON  ARCHES. 

and 

Jf  sB^ttP-sn  for£=f  .....    ^198) 

(b]  Single  Horizontal  Load, 


f  #* 

or 

#;  =  <2(i  -  20/P  +  40^  -  i6/^5)  .....    ^200) 
From  £-(184), 


or 


^(203) 
and 

M^M.+  VJ-Qb  .........    ^204) 

(c]   Temperature. 
From  £-(191), 

^(205) 


i      —  rr-     —7*  -  .... 

/  /   ydx  -   /  y*dx 

J  t/o  t/(,  ^ 


SYMMETRICAL   ARCHES. 


141 


or 


H  = 


^206) 


Given  the  values  of  ffl  and  Vl  for  any  vertical  load  on  the 
left  of  the  crown,  to  determine  Mt,  Mt,  Fa ,  and  //,  for  this 
load,  and  also  for  an  equal  and  symmetrically  placed  load  on  the 
right  of  the  crown. 

Our  formulas  have  been  deduced  for  loads  on  the  left  of 
the  crown,  but  they  are  sufficient  for  the  complete  determina- 
tion of  all  the  outer  forces  for  any  load.  In  fact  we  need  only 
the  values  of  H.  and  Vl  if  graphics  be  employed. 


--i 


FIG.  40. 


In  Fig.  40,  let  P  be  any  load  on  the  left  of  the  crown. 
Make  ab  —  P,  ac  —  V,,  and  dc  =  //",;  then  ad  =  R,  and 
ae  =  R^. 

Through  O  draw  aOe  parallel  to  ae.  From  a,  where  this 
line  cuts  P,  draw  ad  parallel  to  ad.  Then  we  have  the  true 
equilibrium  polygon  for  the  load  P,  from  which  all  the  outer 
forces  can  be  readily  obtained. 

Since  the  arch  is  symmetrical,  evidently  the  values  of  //, , 
Fj,  Ml ,  etc.,  for  the  equal  and  symmetrically  placed  load  P' 
are  equal  to  the  values  of  //, ,  F, ,  M, ,  etc.,  for  the  load  P. 

The  fields  of  loading  which  cause  like  stresses  can  be  found 
in  a  manner  similar  to  that  given  on  page  25  for  arches  having 
two  hinges. 


142 


A    TREA  T1SE   ON  ARCHES. 


THE  THREE-HINGED  ARCH. 

The  three-hinged  arch  as  usually  constructed  is  symmetri- 
cal, and  has  a  hinge  at  each  support  and  one  at  the  crown. 
The  introduction  of  the  third  hinge  materially  simplifies  the 
determination  of  H,  and  //, ;  in  fact  the  problem  is  practically 
one  of  graphic  statics  in  its  simplest  form. 


FIG.  41. 

In  Fig.  41  let  ABC  be  any  arch  having  three  hinges,  and  let 
A,  By  and  C  be  the  location  of  the  hinges ;  then,  evidently, 
there  can  be  no  bending-moments  at  A,  B,  or  C,  and  the  reac- 
tion R\  will  pass  through  A  and  R^  through  C.  For  a  single 
•vertical  load  on  the  left  of  j5.the  reaction  R^  must  also  pass 
through  B,  since  there  can  be  no  moment  at  this  point.  The 
determination  of  //, ,  F, ,  //, ,  and  Fa  now  becomes  quite  sim- 
ple, as  follows : 

Draw  BC  through  B  and  C  until  it  cuts  P  in  D,  and  then 
draw  DA  through  A.  By  simple  resolution  of  the  forces  meet- 
ing in  D  the  values  of  //, ,  Vt ,  etc.,  are  readily  found. 

The  same  results  can  be  obtained  by  applying  the  following 
formulas : 


From  Fig.  41, 


tan  A  =  2"; 


£•(207) 


.  =  (l-  a}  tan  /?,  =  2(1  - 


^208) 


SYMMETRICAL   ARCHES. 


V,  =  P(i  -  k}. 
From  (50),  by  transposition, 


143 
^(209) 

£(210) 


y* 


or 


Pkl 

2f' 


The  stresses  in  the  various  members  of  the  arch  can  now 
be  found  by  the  ordinary  methods. 

The  determination  of  the  fields  of  loading  which  produce 
the  maximum  stresses  has  been  fully  explained  on  page  25  et 
seq. 

The  treatment  of  horizontal  loads  differs  but  little  from  that 
outlined  above. 


FIG.  42. 

Fig.  42  clearly  shows  the  method  for  locating  J?t  and  .#,. 
V,  =  Q-j  and  acts  upward  or  downward  as  Q  acts  towards 
the  left  or  the  right  respectively. 

/ 


.=/-*  tan  0t'=(2/- 


144  A    TREA  TISE   ON  ARCHES. 

From  Fig.  42, 


As  the  Q  loads  are  almost  without  exception  due  to  the 
action  of  wind  and  are  treated  as  static  loads,  the  best  way  to 
obtain  the  stresses  in  the  various  members  of  the  rib  is  to 
determine  the  resultant  values  of  Hl  and  Vl  and  then  treat 
the  problem  graphically. 


CHAPTER  VI. 
COMPARISON   OF   FOUR  TYPES  OF  ARCHES. 

WE  will  take  four  types  of  the  parabolic  arch  having 
EO  cos  (f>  =.  a  constant  and  show  graphically  the  relations 
between  the  values  of  the  outer  forces  for  the  different  types. 

Let  Type  i°  =  arch  with  no  hinges ; 

"  2°  =  "  "  one  hinge ; 
"  3°=  "  "  two  hinges; 
"  4°  =  "  "  three  hinges. 

(a)   VERTICAL  LOADS. 

Comparison  of  H^. 
The  formulas*  are : 


P  I 


See  pages  29,  139,  20,  and  143,  respectively. 

MS 


i46 


A    TREA  TISE    ON  ARCHES. 


These  values  are  represented  graphically  in  Fig.  43,  from 
which  we  see  that  the  2°  type  differs  quite  considerably  from 
the  others,  particularly  for  loads  near  the  crown  and  those  near 
the  springing. 


0.5 

HI 

z 

0.4 

/ 

_1 
2 

I 

S- 

I 

\- 
g 

COMPARISON 
THE  VALUES  C 

OF 
F  H, 

ft 
0.3  £ 

L 

s 

_l 

| 

1 

2 

cc 

J 

o     /•• 

z 

S 

4f 

— 

09 

UI 

1 

0.1  
< 

*/ 

m 

/ 

CO 

> 

^ 

/ 

0 

o 

& 

/ 

0 

0.1 

0.2 

0.3 
V 

0.4 

U.UE 

0.5 

SOP 

0.6 
k 

0.7 

0.8 

0.9  1. 

FIG.  43 

Comparison  of  V^. 
Formulas* — Type  i°.     -W  =  (i  —  >£)s 

2°      ^  =  (i  -  4^) 


See  pages  30,  139,  21,  and  143,  respectively. 


COMPARISON  OF  FOUR    TYPES   OF  ARCHES.  147 

These  values  are  represented  graphically  in  Fig.  44. 


1.0 

NT     - 
0.9\ 

^N 

\ 

> 

M_ 

>"l 

0-6  £ 

CO 

0.5^ 

;N 

3 

>, 

1 

'XjV 

\ 

COR 

1PARISON  OF 

VALUE  qp  v, 

THE 

i 

§ 

A 

s 

0.4  > 

V 

0.3 

\ 

\\ 

0.2 

N\ 

\ 

0.1 

\ 

\ 

\ 

^ 

^^ 

Ov 

0      0 

1      0 

2      0.3      0 

v 

4050 

ALUES  OF 

6      0 

k 

8      0*9      1 

FIG.  44. 
The  equation  for  intermediate  vertical  shear  is 


Vx=Vl  — 


or     Vx  =  KP. 


We  give  below  the  values  of  Vx  for  the  arch  without  hinges 
and  that  with  a  hinge  at  each  support.  The  span  is  divided 
into  twenty  equal  divisions,  and  the  load  P  assumed  to  occupy 
each  point  of  division.  The  values  of  Vx  are  given  for  each 
division. 

These  tables  and  those  given  later  for  maximum  bending- 
moments  are  principally  useful  in  preliminary  computations 
unless  0  is  assumed  to  vary  as  the  secant  of  0,  when  of  course 
they  very  materially  decrease  the  labor  of  calculation. 


148 


A    TREATISE   ON  AKCHES. 


c 

C-.     ---------------- 

e 

Division 
Number. 

H 

Q 

+++++++++  1  1  1  1  1  1  1  1  1  1 

0 

O   O   o"  •"  *•-   cTtnco^-^cor'-iN   «   ^-'o   ET  O   O 

= 

1   1   1   I   !   1   1   1   1  ++++++++++ 

++++++++  1   I   1   1   1   1   1   1   1   l   1 

• 

53"o*~^m£<%S£'SS~~oooo8 

2 

i  l  l  i  i  i  l  l  -H-+++++++++ 

00 

fft!isiiss*iij?ioi§ 

<2 

i  i  i  i  i  i  i  +++++++++    i  i 

++++++  l  i  i  i  i  l  i  ++++++ 

K 

2 

i  i  i  i  i  i  +++++++  i  i  i  i  i  i 

<e 

+  +  +++0    is  ON  ~  CO    i  tcti"i"tttct 

10 

1       1       1       1       1    ++++++    1       1       1       1       1       1       1       1 

M 

itttii  i  L  o^itttctttttt 

to 

r  r  r  r  +++++  r  r  \  \  \  \  \  \  \  \ 

„ 

+++  i  i  i  i  i  +++++++++++ 

CNr-c^vC  «   •^t'^l'vO  coc>w   O   co-l-OOco  w  m 
w   O   "-1   "^0*^^00   w   Of^^-   N   «oorrO^m-^ 

IS 

l  l  i  +++++  l  i  l  i  l  l  l  i  i  i  i 

CO 

++  Mill      +++++++++++ 

co  O^O  oo  co  co  -C         u->  co  Tj-o   ^   O    O*  Tf  w   coco 

eo 

I   l  +++++     l  I  1  l  1  I  I  1  I  il 

M 

||^i|4|||||||tt|tt 

2 

1  +++++  +  1  i   I   1   1  1  1   1   1  1   1   1 

_ 

iHssilltllHIlslll 

o 

-f  ++44+  i  i  i  i  i  i  i  M  i  i  i  i 

|J 

c 

•jaqimiN 

fi°S 

001SIAIQ 

COMPARISON  OF  FOUR    TYPES   OF  ARCHES. 


149 


Division 
Number. 


?%: 


ill  .....  i  +++-1-++++++ 


I   l   I   l   I  I  l   I  l   I  I   I 


++++++  1  1  1  1  1  1  1  1  1  1  1  1  1 

c^,oo   N  mr^io   O   OO   O   OO  r^vnoo   •*  •*•  r-» 
O^OO    I^\O    T  C4    O    TOO    WOO    Tf  Q    r^u^c*^N    »-•    O 


I   i   l   I  I   I   I  I   I     ++++ 

2  £  ?»  S  §,  3  s^  i*  ?  2><i"f> 5  o  q  S  i  1 
r r r r f++ +++++++  r  r  f  r 


i"ti"tioJ>  '  i  o  iitt  i" 

—   t^T^TrvriO    C^cJr^cTQ   rnuito 

«Nc^-*-Tt^cj~-ooqoqq 

1    f  l'    l"  ++++  -f  +-f  l'   \    1    f   1  l 


-f  ++  Mill!  ++++++++++ 

^S  «?^«g>«S5'vg,<2  ^'g  XoT  ?  -  =  8  5"  3  « 

»-p«cn-^c<->N««oOOO<-.--'«OOO 

r  f  r+++4++i'  f  r  r  r  r  r  f  \  \ 


>O   Ooo  m  t^  r^oo  O  T  co 
«««n'«i-c<iN~qqq"-i-i>-«i->i-i«.qq 

r  f ++++++  r  r r f f f f f r \  \ 


+  I    M  ^1^^ 
!'++++++ I    I    I    I    I    I 


i  i 


iooo^iocnr^c^r^  TOO   ~N   l-oo 

++++++ f f  f  r r r  f f f r r r r 


150  A    TREATISE   ON  ARCHES 


Comparison  of  'the  Maximum  Values  of  Mx. 

In  each  of  the  four  types  of  arches  which  we  are  consider- 
ing, if  the  values  of  Ml  ,  F,  ,  and  Hl  be  substituted  in  (41),  we 
find  that 

Mx  =  l-f\K-](Z\ 

where  K  depends  upon  k  =  j  and  Z  upon  z  ==  j  ,  showing  that 

for  parabolic  arches  the  value  of  Mx  varies  with  the  span  alone 
for  given  values  of  k  and  z.     Then  we  may  write 


If  the  values  of  J  be  computed  for  each  load  for  every  value 
of  x  and  tabulated,  the  maximum  values  of  Mx  are  readily 
found  by  taking  the  sum  of  the  values  of  J  having  like  signs. 

We  give  below  the  values  of  J  for  types  i°  and  3°  which 
are  most  common  in  practice. 

It  will  be  noticed  that  the  positive  and  negative  moments 
are  approximately  equal,  although  the  arch  is  divided  into  but 
twenty  equal  divisions.  For  a  uniform  horizontal  load  cover- 
ing the  entire  structure  the  positive  and  negative  moments 
would  be  equal,  since  the  equilibrium  polygon  would  be  a  pa- 
rabola coinciding  with  the  axis  of  the  rib. 

In  Fig.  45  *  is  shown  relatively  the  maximum  values  of  Mx 
for  the  four  types. 

It  appears  from  this  diagram  that  type  i°  has  moments 
which  vary  more  nearly  according  to  the  variation  of  the  sec- 
tion of  the  rib  than  either  of  the  others. 

The  second  type  has  very  large  moments  near  the  springing, 
which  rapidly  decrease  until  about  the  quarter-point,  and  then 

*  This  diagram  is  from  a  note  by  M.  Souleyre:  "Note  sur  1'emploi  de 
quatre  types  d'arcs  dans  les  Fonts,  Viaducts  et  Fermes  Metalliques  de 
grande  portee."  Annales  des  Fonts  et  Chaussees,  mai,  1896. 


COMPARISON  OF  FOUR    TYPES   OF  ARCHES.  I$I 

after  increasing  slightly,  decrease  rapidly,  becoming  zero  at  the 
crown. 

A  crescent-shaped  rib  corresponds  more  nearly  with  the 
variation  of  the  maximum  moments  in  the  third  type. 


FIG.  45. 

As  the  formulas  of  the  fourth  type  do  not  depend  upon  the 
values  of  0.  the  rib  can  be  designed  to  correspond  with  the 
variation  in  the  moments. 

Thus  far  we  have  considered  only  the  live  or  moving  load 
effects. 


152 


A    TREATISE   ON  AXCHES. 


SYMMETRICAL  PARABOLIC  ARCH  WITHOUT  HINGES.* 

PI 
Mx  =  —  _/o  values  of  Jo. 


Point 
of 
Divi- 
sion. 

0 

' 

Crown 
10 

Pom 

-.0789 

+.„„, 

+  .0,35 

+  .0102 

+.0073 

+  0284 

+.0047 

+.0189 

+.0025 
+  .0106 

+.0005 

+    0036 

—  .0010 

-  .0023 

-.003, 

"     3 

4 

HJ354 

-.0647 
-  -°749 

+  .0085 

—  .OJ79 

+  .084- 
+  .0429 

+.0622 
+.1075 

+.0427 
+.0760 

+  .0256 

+.0484 

!:££ 

+.0245 

—  .0011 

+  .0045 

-.0109 
—  .0,16 

-  0239 

'"      I 

-•'°54 
-•°734 

-.0712 
-•0579 

-.03,6 
—  •0357 

—  .0069 

+•0633 

+  .1186 
+  .0705 

+.0793 
+  .119, 

+  .0452 
+  -°744 

+  .0,64 
+  -0363 

-.007, 
+  .0047 

—  .0254 

"      7 

-.0369 

-.0388 

—  .0330 

-.0,94 

+  .0019 

+.03,, 

+.0679 

+  .,126 

+  .0650 

+  .0252 

—  .0069 

"      8 
||      9 

0 

+  .0340 

-.0173 
+  .004, 

-.0,73 
—  .0303 

0 

-.0234 

+.0259 
-.0074 

+  .0605 
+  .0,79 

+  .1037 
+  .0524 

+  .0555 
+  .0965 

+.0,60 
+  •0494 

+  .0625 

+  .0234 

—  .0062 

-!o266 

-•0375 

-.0390 

-.03,3 

-.0,4, 

+  .0,25 

+  .0484 

+  -°937 

•*    ii 

+  .0836 

+  .0388 

+  .0032 

—  .0232 

-.0404 

-.0485 

-•0473 

—  .0370 

—  .0174 

+  .0494 

**      12 

+  •0959 
+  .0994 

+.0490 
+  •0538 

+  .0,08 
+  .0160 

-.0188 

-  .0397 
-.0363 

—  .052, 
-.0509 

-  -0557 
-.0576 

-.0508 
-.0566 

--0372 
-.0478 

-.0,49 

+  .0160 
-.0069 

**      14 

+  .0946 

+  -°534 

+  .0,87 

-.0093 

-  .0307 

-  -0455 

-  -0537 

-  .0552 

-  .0501 

--0385 

-  .0202 

"    J5 

+  .  0820 

+  •0475 

+  .0,83 

-.0056 

-.0242 

-.0376 

-  -0457 

-.0485 

-.046, 

-.0384 

—  .0254 

**     16 

+  .0640 

+  •0,57 

-.0,73 

—  .0280 

-.0348 

-.0379 

—  .037, 

—  .0324 

-.0239 

"     «7 

+  •043° 

1-0259 

+  .01,3 

-.0009 

—  .0,07 

—  .0,80 

-.0229 

-.0254 

-  -0254 

—  .0230 

—  .0182 

"    18 

+  .0225 

.0,37 

+  .  0062 

—  .0001 

—  .0052 

-.009, 

—  .01,7 

-.0132 

-•0134 

-.0124 

—  .0103 

"    »9 

+  .0065 

+  .0040 

+  .00,9 

+  .000, 

-.00,4 

—  .0025 

-.0033 

-.0038 

-.0039 

-.0037 

-.0031 

First  published  by  Prof.  Greene  in  Engineering  News,  vol.  iv. 


SYMMETRICAL  PARABOLIC  ARCH  WITH  Two  HINGES.* 

PI 

MX  =  —/a  values  of /a. 


Point 
of 
Divi- 
sion. 

' 

2 

3 

4 

5 

6 

7 

8 

9 

Crown 
10 

^pni 

+  .0832 

,  .0676 

+  •0523 

+  .0402 

+.0289 

+.0,78 

+.0084 

+  .0003 

-  0066 

—  .0122 

"      3 

"      4 

"      I 

+  .0667 
J.0509 
•°359 

+  .022, 
+  .0097 

+•1359 
+.,05, 
+.0765 
+  .0498 
+  .0257 

+  .1075 
+  .1634 

t32 
+.0480 

+  .08,5 
+  .,250 
+  .1715 
+.12,9 
+  .0767 

+  .0580 
+  .0902 
+  .,260 
+  .1663 
+  .11,8 

+  .0370 
4-.  0590 
+.085, 
+  .1,62 

+.1532 

+  .0784 

t'°3o5 

+.0489 
+.07,7 

+  .,o,6 

+  •0075 
+.0173 
+  .0328 
+•055, 

-.0097 
-  .0006 
+.0155 

—  .0226 
-  .0297 
-.0320 
—  .0283 
—  .0,76 

7 

—  .0013 

+.0043 

+  .0,70 

+.0366 

+  .0632 

+  .0968 

+  .1374 

+  .0849 

+•0394 

+  .00,0 

8 

—  .0,07 

-.0,39 

—  .0097 

+  .0019 

+  .02,0 

+•0475 

+  .0815 

+.1229 

+.07,7 

+.0280 

9 

-.0,83 

-  .0289 

-.03.8 

—  .0270 

-.0,45 

+  .0058 

+  .°338 

+  .0695 

+  •"29 

+.064, 

10 

-.0242 

-.0492 

-.0500 

-.0430 

-.0281 

-.0055 

+  .0250 

+.0633 

+.,094 

ii 

—  .0283 

-lots. 

-.06,8 

-.0670 

-.0644 

—  .0542 

—  .0362 

—  .0105 

+.0229 

+.0641 

*      12 

—  .0307 

-  -0539 

-.0697 

-.0780 

—  .0790 

-.0725 

-  -0585 

-  .037, 

-.0083 

+.2080 

13 

-.0313 

-.0556 

-  .0730 

-.0834 

-  .0868 

-.0832 

—  .0726 

-•0551 

-  -0305 

+  .0010 

J4 

-  .0304 

-•°545 

—  .0720 

-.0833 

-.0882 

-.0868 

-  .0791 

-.0649 

—  -C445 

—  .0176 

'5 

-.0279 

-.0502 

-.0670 

-.078, 

—  .0837 

-.0838 

-  -0783 

-  .0672 

-.0505 

—  .0283 

—  .024, 

-  -0435 

-.0583 

-.0685 

-  .0740 

-.0749 

-.07,1 

—  .0627 

-  -0497 

-  .0320 

*7 

-.0191 

—  .0347 

-  .0467 

-.0550 

-.0598 

-.0609 

-  -0585 

-  -0525 

-  .0429 

-  .0297 

18 

-.0133 

-.024, 

-  -0325 

—  .0385 

-  .0420 

-.0430 

—  .04,6 

-.0365 

—  -0314 

"    19 

-.0068 

—  .0167 

—  .0198 

—  .0210 

-  .0197 

-  .0,66 

—  .0,22 

First  published  by  Prof.  Greene  in  Engineering  News,  vol.  iv. 


COMPARISON  OF  FOUR    TYPES  OF  ARCHES.  153 

The  dead  load  is  very  nearly  a  uniform  horizontally  distrib- 
uted load,  and  hence  the  moments  due  to  this  load  are  prac- 
tically zero  in  the  four  types. 

If  in  the  i°  and  2°  types  the  dead-load  stresses  are  com- 
puted as  if  the  ribs  were  hinged  at  the  springing  and  then  built 
with  hinges  at  the  springing,  when  the  falseworks  are  removed 
the  rib  will  settle  into  position  and  the  dead-load  stresses  will 
be  practically  those  computed. 

From  Fig.  45  we  see  that  the  live-load  flange-stresses  are  a 
minimum  for  the  i°  type,  or  the  arch  without  hinges.  A  rib  con- 
structed with  pins  at  the  springing  is  very  easily  made  into  a 
rib  with  fixed  ends  by  arranging  the  details  so  that  the  flanges 
may  be  rigidly  connected  with  the  piers  or  abutments  after 
the  falseworks  are  removed. 

This  method  is  followed  by  French  and  German  engineers 
in  many  cases,  especially  for  masonry  arches  and  metal  arches 
with  solid  webs. 

The  arch  without  hinges,  or  type  i°,  appears  to  be  the  most 
economical  of  the  four  for  the  dead  and  live  loads. 

There  remains  to  be  considered  the  effect  of  temperature. 

Comparison  of  Temperature  Effects. 
Typei0.     H^Acf.     M, 


2°.     H,  =        Aff°-     M,  =  HJ. 

3°.     H,=   l^Aet\     M>=o. 
4°.    H,  =  o.  M,  =  o. 

Hence  for 

Type  i°.     Mx  =  //,(£/-  y)  =       Aet\^f  -  6y}. 


154  A    TREATISE   ON  ARCHES. 

Type  2°.     Mx  =  Htf  -  y)  = 


From  which  we  see  that  the  effect  of  temperature  is  great- 
est in  the  i°  type  and  least  in  the  4°  type;  also  that  the  effect 
in  the  2°  type  is  greater  than  that  in  the  3°  type. 

For  structures  carrying  moving  loads  the  second  and  fourth 
types  are  not  desirable  on  account  of  vertical  vibration  of  the 


MAXIMUM  MOMENTS 
DUE  TO  THE  WIND 
BLOWING  AGAINST 
ONE  SIDE  OF  THE 
ARCH  WITH  A  FORCE 
OFpPER  UNIT  OF 
THE  RISE/ 


FIG.  46. 

structure,  leaving  the  first  and  third  types  to  be  selected  from. 
Complete  calculations  show  that  for  large  structures  the  first 
type  is  more  economical  as  well  as  mor?  rigid,  and  practice  proves 
this  type  well  adapted  to  the  work  required  for  a  railway 
bridge. 


COMPARISON  OF  FOUR    TYPES  OF  ARCHES.  155 

(V)   HORIZONTAL  LOADS. 

Horizontal  loads,  being  usually  due  to  wind,  may  be  consid- 
ered as  a  dead  load,  covering  the  arch  on  one  side  from  the 
crown  to  the  springing. 

For  a  uniform  load/  per  unit  of  height  of  the  arch,*  Fig. 
46  shows  the  relative  values  of  max  Mx  for  the  four  types. 

Here  we  see  that  the  first  type  more  nearly  agrees  with  the 
variation  of  0  in  the  variation  of  the  moments,  so  that  the 
conclusion  drawn  above  remains  unchanged  for  structures 
carrying  a  moving  load.  In  case  there  is  no  moving  load,  as 
in  roof-trusses,  the  fourth  type  appears  to  be  most  economi- 
cal. This  type  is  almost  always  employed  by  American 
engineers  for  large  roof-trusses. 

Comparison  of  Types  i°,  3°,  and  4°  designed  for  a  Single-track 
Railway  Bridge  having  a  Span  0/416  Feet.\ 

To  more  clearly  show  the  relation  between  the  three  types 
1°,  3°,  and  4°,  a  comparison  of  the  maximum  stresses  in  the 
individual  members  of  a  trussed  parabolic  arch  rib  are  shown 
in  Figs.  47,  48,  49,  and  50. 

The  diagrams  show  the  maximum  stresses  due  to  dead 
load,  live  load,  wind,  and  changes  in  temperature. 

Figs.  47  and  48  clearly  indicate  the  superiority  of  the  arch 
without  hinges  for  economy  in  the  flanges. 

Figs.  49  and  50  show  that  there  is  little  choice  between  the 
types  as  far  as  the  web  is  concerned,  there  being  a  remarkably 
close  agreement  between  the  stresses  for  the  three  types. 

The  principal  data  employed  are  as  follows  (see  Fig.  51) : 

Span 416'  o" 

Rise   67'  o" 

*See  note  under  Fig.  45. 

f  The  computations  for  this  comparison  were  made  by  Messrs.  Crockwell, 
Wiggins,  and  Shaneberger  in  connection  with  their  theses  for  graduation 
from  the  Rose  Polytechnic  Institute. 


T56 


A    TREATISE   ON  ARCHES. 


CO       , 

Q      \ 

oc       V 
O 

X 

o 

o 

I- 
o 


5JL 


NOISN31 


COMPARISON  OF  FOUR    TYPES   OF  ARCHES.  1 57 


15°  A     TREATISE    ON  ARCHE$. 

Batter  of  arch  planes. I  in  3 

Depth  of  rib  at  crown 6'  o" 

"       "     "    "  skewbacks 10'  O77 

Moving  load  per  lineal  foot  ot  span 4000  Ibs. 

Dead         "        "       "         "      "  superstructure 1500   " 

"            "       "       "         "      "  arch' 1000   " 

Wind        "       "       "         "      "  span  (live).    300   " 

"     "      "     (dead) 600   " 

Range  of  temperature ±  80°  F. 


FIG.  51. 

Relative  Weights  of  Steel  in  One  Arch  Rib,  including  Gusset- 
plates,  Rivets,  etc. 

Type  i °,  without  hinges l.oo 

"      i.2i 


3  i  two 

4°,  three 


•30 


CHAPTER  VII. 
APPLICATIONS. 

IN  the  preceding  pages  we  have  deduced  formulas  for  deter- 
mining the  various  reactions  and  moments  which  result  from  the 
application  of  vertical  and  horizontal  forces  to  the  linear  elastic 
arch,  that  is,  we  have  assumed  that  the  forces  were  applied 
upon  the  central  line  or  neutral  axis  of  the  arch  rib.  In  practice 
this  evidently  is  not  always  the  case,  especially  where  a  super- 
structure is  supported  by  arch  ribs  having  considerable  depth. 

The  weight  of  the  arch  rib  alone  may  without  serious  error 
be  assumed  as  applied  to  the  centre  line  or  neutral  axis. 

Vertical  Loads. — Vertical  forces  due  to  the  superstructure 
and  moving  loads  may  be  assumed  to  act  where  they  intersect 
the  neutral  axis  in  flat  ribs  and  in  trussed  ribs  where  one 
system  of  the  web  bracing  is  vertical*  The  same  assumption 
may  be  made  for  plate-girder  ribs,  as  they  are  either  very  shal- 
low, as  in  bridges  of  short  spans,  or  the  forces  due  to  the 
superstructure  are  applied  to  the  rib  quite  close  together. 

For  the  condition  where  a  vertical  force  does  not  intersect 
the  neutral  axis  of  the  rib,  as  in  the  case  of  a  large  semicircular 
rib  near  the  supports,  the  following  method 
may  be  employed.  In  Fig.  52  let  P  be  a 
vertical  force  applied  at  B  which  does  not 
intersect  the  neutral  axis.  At  the  centre 
of  the  strut  BD  place  the  two  equal  and 
opposite  forces  P\  then  we  have  for  the 
equivalent  of  the  force  P  applied  at  B 
the  force  P  applied  at  C  and  the  couple  Pd. 
The  reactions  can  now  be  found  by  apply- 
ing the  formula  for  a  vertical  load  and  that  for  a  couple.  In 
passing  we  may  say  that  this  method  is  general  and  can  be  ap- 
*  See  Fig.  51. 

159 


100  A    TREA  TISE   OX  ARCHES. 

plied  for  any  load  whether  its  direction  intersects  the  neutral 
axis  or  not. 

Horizontal  Loads. — Horizontal  forces  in  the  plane  of  the 
arch-rib  seldom  occur  in  practice,  excepting  in  the  case  where 
the  arch  is  employed  for  supporting  a  large  roof.  In  this  case 
the  horizontal  force  is  the  horizontal  component  of  the  wind 
load. 

If  the  stresses  due  to  the  wind  are  small  in  comparison  with 
those  caused  by  the  total  dead  weight  of  the  structure,  the 
wind  forces  may  be  assumed  to  act  upon  the  neutral  axis  where 
the  normal  components  intersect  it  in  the  determination  of  re- 
actions, etc.  If  greater  accuracy  is  desired,  then  the  force 
may  be  replaced  by  an  equal  force  and  a  couple,  as  explained 
above  for  vertical  forces. 

Wind  Loads. — We  have  just  explained  how  to  consider 
wind  loads  in  the  plane  of  the  arch.  There  remains  to  be 
discussed  the  action  of  the  wind  against  the  arch  and  super- 
structure perpendicular  to  their  plane. 

The  superstructure  is  usually  composed  of  a  roadway  sup- 
ported by  columns  or  towers  according  to  the  magnitude  and 
design  of  the  structure. 

The  action  of  the  wind  against  the  roadway  creates  a  hori- 
zontal reaction  at  the  top  of  each  post  or  tower.  This  reaction 
is  transmitted  to  the  arch-rib  in  the  form  of  an  equal  horizontal 
force  at  the  foot  of  the  column  or  tower,  and  a  couple  which 
is  equivalent  to  a  vertical  force  acting  upward  on  the  wind  side 
of  the  structure  and  an  equal  vertical  force  acting  downward 
on  the  opposite  side  as  illustrated  in  Fig.  53.  The  vertical 
T  -T-  forces  are  treated  as  explained  above. 

—  The  horizontal  force  with  that  due  to 
the  direct  action  of  the  wind  against  the 
rib  must  be  considered  differently.     The 

—  actual  action  of  these  forces  is  very  com- 
plex unless  we  make  the  assumption  that 
the  arch-ribs  act  a?  the  chords  of  a  canti- 
levered  beam  having  a  length  equal  to 
one    naif   the  length  of  the  axis  of  the 

arch.     Under  this    assumption   the   lateral   systems  may   be 


1 


APPLICATIONS.  l6l 

developed  and  the  stresses  in  the  different  members  found  by 
ordinary  methods.  Although  this  method  is  not  correct,  yet 
its  simplicity  and  probable  safety  commend  its  use. 

Maximum  Stresses. — We  have  explained  in  Chapter  II  the 
methods  for  selecting  those  forces  which  cause  the  maximum 
shears  and  moments  at  any  point.  Another  method  may  be 
employed  which  has  many  features  in  its  favor.  The  values 
of  the  reactions,  etc.,  may  be  found  for  each  load,  and  then 
the  stresses  in  each  member  of  the  rib  due  to  each  individual 
load.  These  stresses  being  tabulated,  the  maximum  positive 
and  negative  stresses  are  readily  determined  by  simple  addi- 
tion. This  method  is  long,  but  has  the  advantage  of  being 
free  from  errors,  and  if  each  load  is  taken  as  unity,  the  stresses 
obtained  for  the  individual  loads  will  be  coefficients  which  can 
be  applied  to  any  load.  The  latter  feature  is  of  considerable  im- 
portance, as  very  often  the  magnitudes  of  the  loads  are  changed 
before  the  final  computation  is  made.  In  very  large  structures 
where  the  moving  load  is  small  in  comparison  with  the  dead 
load  it  is  customary  to  make  but  two  computations  for  the  mov- 
ing load  :  one  for  the  moving  load  covering  the  entire  structure, 
and  a  second  for  the  load  covering  one  half  of  the  span. 

Character  of  Reactions. — In  the  hinged  or  fixed  arch  the 
-vertical  reactions  (F,  and  F,)  always  act  upivard when  the  verti- 
cal forces  which  are  applied  to  the  arch  act  downward,  and  the 
horizontal  reactions  (//,  and  H^)  act  from  the  supports  towards 
the  centre  of  the  span.  In  case  the  vertical  forces  act  upward, 
F,  and  F,  act  downward  and  //,  and  fft  act  away  from  the 
centre  of  the  span. 

In  the  case  of  horizontal  loads,  if  the  load  acts  from  the 
left  towards  the  right,  Fj  acts  downward  and  F,  acts  upward. 
Both  of  the  horizontal  reactions  act  from  the  right  towards 
the  left. 

Co-ordinates  y9,  x^y^x^y^x%. 

Vertical  Loads. — The  ordinate  y^  is  always  measured  upward 
from  the  long  chord  of  the  arch. 

The  ordinate  yl  is  always  measured  upward  at  the  left 


162  A    TREA  TISE   ON  ARCHES. 

support  for  loads  on  the  right  of  the  crown.  For  loads  adja- 
cent to  the  left  support  yl  is  measured  dowmvard.  y1  is  zero 
for  a  load  near  a  point  which  is  four  tenths  the  span  from  the 
left  support,  this  distance  varying  with  different  arches. 

The  abscissa  xl  is  measured  to  the  left  of  the  left  support 
when  yl  is  measured  ztpward,  and  to  the  right  when  y1  is  meas- 
ured downward. 

xl  and  yl  are  zero  when  the  arch  has  a  hinge  at  the  left  sup- 
port. 

The  directions  of  x,  and  y,  are  easily  determined  from  what 
has  been  said  concerning  x^  and_y,. 

Horizontal  Loads. — xa  is  always  measured  towards  the  right 
from  the  left  support  for  loads  on  the  left  of  the  crown. 

yl  and  j/,  are  always  measured  upward  at  the  left  and  right 
support  respectively. 

xl  is  always  measured  to  the  left  of  the  left  support,  and 
x^  to  the  right  of  the  right  support. 

Bending  Moments  at  the  Supports. — For  arches  with  hinges 
at  the  supports  Ml  and  Mt  are  zero. 

When  j/,  is  zero  Ml  is  also  zero.  When  the  extremity  of  jy, 
lies  between  the  flanges  of  the  arch-rib  both  flanges  have  the 
same  kind  of  stress ;  for  vertical  loads  acting  downward  this 
stress  is  compression. 

When  the  extremity  of  j,  lies  above  the  rib  the  upper  flange 
is  in  compression  and  the  lower  in  tension  for  vertical  loads  act- 
ing downward,  or  Mi  is  positive.  When  yl  is  measured  down- 
ward Ml  is  negative  and  the  upper  flange  is  in  tension  and  the 
lower  in  compression  unless  the  extremity  of  yl  falls  between 
the  flanges,  when  both  are  in  compression  for  vertical  loads 
acting  downward. 

To  illustrate  the  application  of  our  formulas  we  will  now 
solve  various  examples  in  detail. 

i°.  Given  a  parabolic  arch,  with  a  hinge  at  each  support, 
having  a  span  of  100  and  a  rise  of  25,  determine  H,  for  a  load 
P  placed  at  a  distance  25  from  the  left  support. 

Here          /  =  100,    7=25,     and     £  =  0.25. 


A  P PLICA  TIONS.  1 63 

From  (64.0)  we  have 


The  vertical  reaction  Vl  is  found  from  (65),  or 

F,  =  (i  -  o.25)/>  =  0.75^. 
From  (66#)  we  have 

7.  =  25(1.3474)  =  33-68. 

In  a  like  manner  the  values  of  Hl ,  F, ,  and  j0  can  be  found 
for  any  other  vertical  load. 

The  method  employed  above  was  the  common  method 
neglecting  the  effect  of  the  axial  stress.  Although  this  is  of 
little  consequence. in  this  case  (see  Appendix  C),  we  will,  how- 
ever, give  the  solution  which  includes  the  axial  stress. 

For  this  we  need  the  values  of 

/* 

m  =  (the  radius  of  gyration)*,    /  —  parameter  =  — :,  and  00. 

Let  m  be  assumed  =  4  (an  average  value), 
p  =  50     and     00  =  0.7854. 
Then,  from  (74), 

H 15 \  8  X  100(25)* 

1       8  X  100(25)'  +  30  X  4  X  50  X  0.7854!  15  '' 

4(100)'  ) 

2(50+ so)"1  ?.P 

or 

H,  =  0.0000297133333$,  -  2QOPk(l  -  k)\, 
which  is  general  for  this  particular  arch. 


164  A    TREA  TISE   ON  ARCHES. 

Substituting  the  values  of  ^,  and  k,  we  have 

//,  =  0.000297!  18559.8  -  37.5}  =  0.550P. 

From  the  approximate  equation  (75), 
HI  =  0.5568(0.9885)  =  0.5  $oP, 

the  difference  in  results  being  in  the  fourth  decimal  place. 
The  value  of  V^  remains  unaffected  by  the  axial  stress. 
From  (76), 


By  the  common  method  yn  =  33.68,  which  is  but  0.41  less 
than  obtained  above. 

In  a  similar  manner  any  other  vertical  load  may  be  treated. 

2°.  Let  a  horizontal  load  Q  be  applied  in  place  of  the  ver- 
tical load  P.  Then,  by  the  common  method  from  (77)  or  (77#), 

H,  -  0.57420. 

Note  that  the  values  of  fft  are  given  by  Table  III  when 
Q  =  unity. 
From  (780), 

V,  =  4  X  0.25  X  0.18750  =  0.1875^. 

From  (79), 

*0  =  o.5742/  =  57.42. 

If  the  axial  stress  is  included  in  our  calculations  we  have 
to  apply  (83),  which  contains  the  factor  -.  But 


APPL1CA  TJONS.  165 

=  0.4949; 


mp(a  +  0.)  _  4  X  50(0463  +  0-785)  _ 

—   —  —  0.0074. 


Hence 

//,  =  0.4949!  2(0.5742)}  (2  +  0.00740  =  0.57570, 

which  is  but  a  very  small  amount  larger  than  the  result  found 
by  the  common  method. 

F,  =  0.18750,  as  before. 

From  (85), 


3°.  In  place  of  the  loads  P  and  Q,  suppose  the  arch-rib 
constructed  of  metal  having  a  modulus  of  elasticity  E  = 
28,000,000,  and  let  the  temperature  rise  50°.  What  will  be  the 
value  of  //,  if  the  coefficient  of  expansion  of  the  metal  is 
0.0000055? 

From  (86), 

„,  =  15^^(0.0000055,50. 

If  6  is  taken  at  the  crown,  cos  0  =  i. 
Let  0  =  4  ;  then 

H,  =  92.4. 
From  (87),  which  includes  the  axial  stress, 


Since  a  rise  in  temperature  tends  to  lengthen  the  arch  rib, 


166 


A    TREATISE   ON  ARCHES. 


the  span  will  tend  to  increase,  hence  Hl  must  act  from  left 
towards  the  right. 

4°.  Let  the  arch  be  assumed  parabolic  in  shape  and  fixed  at 
the  ends.  Let  a  load  P  be  applied  at  the  quarter-point  and 
determine  the  reactions,  etc. 

The  following  data  will  be  used  : 

/  =  100,        /=  25,         00  =  0.7854,         a=  0.463. 
For  the  value  of  Hl  we  have,  from  (91)  or  (910), 

HI  =  -V5-  W(o-0350^=  0.5265^. 
From  (92)  or  (92^)  we  have 

M,  =  i§*  (  -  o.io54)/>  =  -  5.27^. 
From  (93)  or  (930), 

V,  =  o.8437/>. 
From  (92)  or  (920), 

Mt  =  -4*(  +  0.0820)7*  ^  +4.IO/5. 
From  (93)  or  (93^),  letting  k  —  i  —  k  —  0.75, 


From  (94), 

}'o  =  72S  —  3°>  measured  up. 

From  (95), 

yt  =  —  0.4(25)  =  —  10,  measured  down. 

From  (96), 

5=  +7777.  measured  up. 


APPLICATIONS.  167 

From  (97)  or  (970), 

x,  =  —  \°(4(— 0.625)  =  +6.2 5,  measured  to  the  right. 
From  (98)  or  (98^), 

*a  =  —  ^V0-  (2-625)  =  —  26.25,  measured  to  the  right. 

A  good  check  upon  the  above  work  is  to  lay  off  the  ordi 
nates  and  see  if  the  two  reaction  lines  meet  on  the  load  line  P 
as  indicated  in  the  figure  below. 


Thus  far  the  formulas  of  the  common  method  have  been 
employed.  We  will  now  consider  the  effect  of  the  axial  stress. 

For  this  case  we  apply  (101)  to  obtain  the  value  of  ffl , 
letting  m  =  4  and/  =  50.  From  (102), 


15  X  ioo  X  25 


4  X  ioo  X  625  +  90  X  4  X  50  X  0.7854 


=  0.1419; 


2/1/+2/)  50(100) 


^Q 


Then 


,  =  0.1419}  100(0.0351)  —  0.24(0. 1875) * 


H,  =  0.49 1  P. 
By  the  approximate  formula  (103), 

HI  —  0.935(0.5265)7'  =  0.492/1 


168  A    TREA  TISE   ON  ARCHES. 

For  the  bending-moment  J/a  we  employ  (107),  in  which  there 
are  several  coefficients  which  are  constant  for  this  arch.  We 
will  first  compute  these: 


/- 

=  ° 

00  =  0.7854.         at  =  0.463. 
Then,  from  (107) 

^,(0.4805)  =  ^|i4.98l  -  99-025^(1 


+  0.2377/^(1  -  K)k  -  o.6/?{o.7854(2/^  -  0  +  0.463}. 

In  the  case  we  are  considering  //,  =  o.49i/'and  ^  =  0.25. 
Making  the  proper  substitutions,  we  have 

^(0.4805)  =  +  7-355^'-  22.O53/3 
+  16.405^ 
+  0.044^—  0.043/3 
or 


4  P  PLICA  T10NS  1  69 

To  determine  the  value  of  M,  we  substitute  i  —  k  for  k  in 
the  above  formula,  or 

^1(0-4805)  =  +  7.355^       22.053^ 

+  n.7i5^ 

+  o.044/>  -  0.043  P 

or 


As  a  check  we  will  apply  (112)  to  this  case  ;  then 

M>  =  -    3-555^  - 

—  22.270/>4-     O.I 

+   0.047/3 
or 

M,  =  -  5.953^ 

(6.1  16  —  5.953)^  =  o.i6P,  or  an  error  of  about  2#,  caused  by 
neglecting  decimals. 
From  (113), 


or 

V,  =  0.8467/5, 

which  differs  but  o.oo^P  from  the  value  obtained  by  the  com 
mon  method. 
From  (51), 


or 


-6.116 

yl  = =  —  12.46,  measured  downward, 

0.491 


A     TREATISE   ON  ARCHES. 


and 


y  =    '    3'555  =  _|_  7.24,  measured  upward. 
0.491 


From  (54), 
Therefore 


6.116 


0.8467 


=  7.2,  measured  to  the  right, 


and 


3-555 


i  —  0.8467 


=  23.2,  measured  to  the  right. 


From  (50), 


MA.  ya      —6.116  +  21.18 

*  =  —ffT  = 5^T-  -  =  3°-6' 


To  show  the  effect  of  the  axial  stress  upon  each  of  the 
quantities  we  will  tabulate  our  results : 

COMPARISON  OF  RESULTS. 


Function. 

Common  Method. 

Exact  Method. 

Difference. 

Percentage  of 
Common  Method. 

H, 

0.5265^ 

0.491^ 

0.035JP 

6.6 

/A 

0.5265  P 

0.49I/' 

o.035/> 

6.6 

r, 

0.8437P 

0.8467  P 

0.003  P 

0-3 

T, 

O.l&lP 

O.I533-P 

0.003  / 

2.0 

Mi 

5.27-P 

6.H6P 

0.84/1 

16.0 

M, 

4.ioP 

3-555^ 

0.55^ 

15.5 

y<i 

30.0 

30.6 

0.6 

2.O 

yi 

IO.O 

12.46 

2.46 

24.6 

y% 

7-77 

7.24 

0-53 

7-0 

*l 

6.25 

7-2 

0.95 

15-2 

x* 

26.25 

23.2 

3.05 

II.  6 

A  P PLICA  T1ONS.  1 7  I 

5°.  Let  the  vertical  load  be  replaced  by  a  horizontal  load 
acting  from  left  to  right. 

For  the  value  of  Hl  by  the  common  method  we  have,  from 
(115)  or  (i  16), 

HI  =  0.63290  acting  towards  the  left. 

From  (117)  or  (118), 

Ml  =  25(0.2  109)0  =  5.272(2. 

Now  since  Q  acts  towards  the  right,  Ml  will  be  negative. 
From  (119)  or  (120), 

M*  -  -  25(0.1171)0  =  -  2.9270. 

Since  Q  acts  towards  the  right  and  our  formula  was  deduced 
for  Q  acting  towards  the  left,  Mt  will  be  positive. 
From  (121)  or  (i2ia), 

V^  =  12^(0.0351)0=  0.10530,  acting  downward  ; 

V^  =O.IO530,  acting  upward. 
From  (123)  or  (123^), 

ji  =  25(0  3332)  =  8.33,  measured  upward. 
From  (124)  or  (1240), 

y^  =  25(0.3189)  =  7.97,  measured  upward. 
From  (125)  or  (125^), 

x^  —  100(0.5)  =  50,  measured  to  the  left. 
From  (126)  or  (126^), 

xt  =  100(0.2778)  =  27.78,  measured  to  the  right. 


1 72  A    TREATISE   ON  ARCHES. 

From  (127)  or  (1270), 


0  =  1.25  =  62.5,  measured  to  the  right. 


Note  that  for  horizontal  loads  xl  and  ^s.are  always  meas- 
ured outward  and  y^  and  y^  upward,  without  regard  to  the  di- 
rection of  Q. 

We  will  now  consider  the  effect  of  the  axial  stress. 

The  value  of  //,  is  given  by  (128)  or  (130),  which  contain 
several  constants,  which  we  will  compute  first. 


C  —  O.I4I9,  (a  +   00)   =    1.248,  --   =  0.24. 

Then,  from  (130), 


,  +  o.24(i.248)(2| 
or 

Ht  =  0.141914.219  -j-  0.299}  Q  =  0.641  1  Q. 

The  value  of  M,  is  given  by  (132). 
D  =  0.99025. 


0  _ 

Y/-/+  —rD  =  14-98- 


00    =  0.299. 
Then  (132)  becomes 

,  -  0.1715^(1  -  k) 


—  0.2 


A  P  PLICA  TIONS.  I  73 

or 

^,(0.4805)  =  +    9-6040  -    0.0320 

-{-  12.012(2  —  22. 


hence 


As  in  the  common  method,  Mt  is  positive,  since  Q  acts  from 
left  to  right. 

By  making  k  equal  I  —  k  and  substituting  Ht  for  H  the 
value  of  Ml  becomes 

^(0.4805)  =  +  5-376(2  -    0.032(2 

+  8.8870-  16.865  (2 

-    0.2980. 

Therefore 


From  (i37)» 


or 

Ft  =  0.0960(2. 

From  (51), 

«  —  __l  -  =  9.5,  measured  upward, 


0.64  II 
and 


v  =    3-°3    _  g  4  measured  upward. 
7«       0.3589 


174  A    TXEATISE   ON  AKCHES 

From  (54), 

^,  =  — — g  =  63.8,  measured  to  the  left, 
and 

^a  =  =31.6,  measured  to  the  right. 

From  (58), 

1 106.2  —  606.  i 
•*„  =  — —  =  02.  i,  measured  to  the  right. 

6°.  For  temperature  stresses  let  the  arch  be  of  metal,  having 
a  modulus  of  elasticity  E  =  28,000,000  and  a  coefficient  of 
expansion  e  =  0.0000055.  Assume  the  temperature  to  rise  50°. 

From  (140),  neglecting  the  effect  of  the  axial  stress  we  have 


_ 

~^T~ 
Let  6  cos  0  =  4;  then 

H,  =  554-4- 
From  (144), 

M,  =  M,  =  554-4-8/  =  9240. 
From  (145), 


If  the  effect  of  the  axial  stress  is  included,  we  have,  from 
(139), 

//,  =  (>  1419(3  X  EO  cos  (f>et°}l  =  525. 
From  (141), 


^(0.4805)  =  Ji  14.9$}  Aet°  -  o.uSSAet0. 


APPLICATIONS.  175 

Now        Aet°  =  28000000  X  4  X  50  X  0.0000055 

or         Aet°  =  30800. 
Therefore 


From  (51), 


H,          525 


8744        ^ 

=  IO.O. 


7°.  Let  the  arch  be  loaded  from  the  left  support  to  the 
crown  with  a  uniform  horizontally  distributed  load;  then 
k"  =  o  and  k'  =  0.5. 

From  (147), 


From  (148), 


From  (149)* 


From  (150), 


The   location    of   the   point   where    the    true   equilibrium 
polygon  starts  can  be  found  from  (51). 


176  A    TREATISE   ON  ARCHES. 

1562 

y.  =  -  =  6.2.  measured  downward  * 
25 

1562 
y^  =  --  =  6.2,  measured  upward. 

8°.  What  will  be  the  vertical  deflection  of  the  arch  at  the 
crown  when  there  are  two  equal  and  symmetrical  loads  placed 
at  the  quarter  points? 

Let  E  —  28000000  and  0  cos  0  =  4; 


Since  the  arch  is  fixed  at  the  ends,  J00  =  o;  then  ^(84), 
page  55,  becomes,  making  x  =  1/2, 


(We  have  neglected  the  effect  of  the  axial  stress.) 
—  -T  =  0.0000037  about. 


—   3.  5  iP  +  50.000^ 


-  19.74^ 


A  P PLICA  7VOJVS.  177 

Therefore 

6y  =  0.0000037(0.76)0  =  0.00000280. 

Suppose  Q  =  30000,  then  dy  =  0.084  J  and  if  our  span  is 
measured  in  feet 

Sy  =  1.008  inches. 

The  sign  being  positive  indicates  that  the  crown  rises 
under  the  action  of  these  two  loads  placed  at  the  quarter 
points. 

Thus  far  we  have  considered  only  parabolic  arches.  We 
will  now  solve  a  few  similar  problems  for  a  circular  arch  having 
a  span  of  100  and  a  rise  of  25.  The  following  data  will  be 
employed  : 

/=  100,  /=25,        ^=62.5, 

r\ 

k>  =  37-5»    0o  =  53°  7i'»  **  =  j^  —  0.00102, 

£  =  25,          a  =  23°  35'. 

9°.  Determine  //, ,  Vl ,  etc.,  by  the  common  method,  as- 
suming the  arch  to  have  a  hinge  at  each  support.  Vertical 
load  P. 

From  (i6oz), 


In  order  to  use  Table  XVII  we  must  determine  the  values 

,  2(/>0        ,    a 
of  — —  and  — . 


^  =  0.590,    ~  =  0.443. 

it  <P<, 

Entering  Table  XVII  with  these  values,  we  have  by  inter- 

A 

polation  —  =  0.570.     Hence 

H^  =  0. 


[?8  A    TREATISE   ON  ARCHES. 

From  (161), 

V,  =  0.750/3. 
From  (163), 


From  (164), 


'(1.407) 

v.  i  J3 

or 


F;  =  0.75  P,  as  before. 
From  (50), 


From  the  approximate  equation,  #i  =  fj,(i  —  *),  as  ex- 
plained in  Appendix  C,  the  value  of  Ht  becomes 

H,  =  0.57(0.9885)^  =  0.5634^, 

no  change  in  the  figures  occurring  until  the  fourth  decimal  is 
reached. 

10°.  Let  the  vertical  local  P  be  replaced  by  a  horizontal 
load  Q  acting  towards  the  left.  Then  we  have  to  employ  for- 
mula (172),  if  the  axial  stress  is  neglected. 

a  —  sin  a  cos  «•  =  /?,„     =  0.04491 ; 

sin  <x  —  a  cos  a  =  44lt  =  0.02288 ; 

2  cos  00  =  1.2; 

0n  —  3  sin  00  cos  00  -f  200  cos'  00  =  J18  =  0.155. 


or 


APPLICATIONS.  179 

Then 

H  =  l-Q  I     4-  °-°449I  —  1.2(0.02288)  \ 
1       2*{  0.155  ) 


From  (176), 

From  (178), 

or 

.*•„  —  0.799!  1. 1 125  }62. 5  =  55.55,  measured  to  the  right. 
From  (177), 

*.  =  ^=55.6*. 

The  difference  in  these  values  is  due  to  the  omission  of 
decimal  figures. 

If  the  axial  stress  is  not  neglected  the  operation  becomes 
considerably  longer.  The  formula  to  be  employed  is  (180)  or 
(181). 

All  but  two  of  the  terms  in  (180)  have  been  evaluated 
above. 

0o  +  sin  0o  cos  00  =  Alg  =  1.4073, 
«+  sin  a  cos  a—  A19  =  0.7782. 

Then 

(  o.i 55+  0.0449  -0.01745      \ 

ffl  =  Q 1  +  0.00102(1.4073  +0.7782)  v 

(    0.310  +  0.00204(1.4073)     ) 

or 

H,  =  0.558(2. 


ISO  A    TREATISE   ON  ARCHES. 

As  before, 

F;= 
Then,  from  (183), 


ii°.  Let  the  arch  be  fixed  at  the  ends,  and  let  a  load  Pbe 
placed  at  the  quarter-point  on  the  left  of  the  crown  ;  then  by 
the  common  method  //,  will  be  found  from  (192). 

The  following  quantities  will  be  required  : 

2  sin  00  =  i  .60, 
cos  #4-  «sin  a=  JM  =  1.081 1, 
sin  00[2  cos  0o  4-  0o  sin  00]  =  JSI  =  1.5535, 
0o*  4~  0o  sin  0o  cos  00  —  2  sin"  00  =  ^ao  =  0.0250. 

Hence 

'     '  -  1-5535  - 


or 

fft  =  0.546^. 

The  value  of  Mt  is  given  by  (196),  in  which  the  following 
terms  appear : 

200  =  1.862,      sin  a  =  0.400,     cos  a—  0.9164,     sin  00  =  0.800, 

cos  0o  =  0.600. 

0o  4~  sin  00  cos  0o  =  4lt  =  1.4073, 
—  00  4-  sin  0o  cos  0o  =  —  /?,„  =  —  0.4474, 

cos«4~  a  sm  a  =  dn  =  1.081 1, 
cos  0,  4-  0o  sin  0»  =  JM  =  1.3407, 
sin  00  —  0o  cos  0o  =  ^^,,  =  .2437. 
Then 

0.546(62.5)  62.5 

0.93 1      l°'2437J  ^  r  i.862(  -  0.4474) 
10.4(0.93  i)[(o.9i64)(o.8)  -  1.4073]  4-  (0.40(0.93 i)(o.8) 
4-  (-  o.4474)[i.o8ii  -  1.3407]}, 


APPLICATIONS.  1  81 

which  reduces  to 


We  see  from  this  problem  that  the  solutions  of  the  formulas 
for  the  fixed  circular  arch  are  considerably  longer  than  those 
for  the  parabolic  arch,  even  with  the  aid  of  the  Tables. 

In  practice  it  will  be  found  that  if  Hl  ,  M,  ,  etc.,  are  deter- 
mined for  loads  placed  so  that  a  will  be  in  even  degrees,  and 
then  the  values  of  //,  and  J/,  interpolated  from  a  diagram, 
much  time  will  be  saved.  This  method  also  is  less  liable  to 
have  errors  in  individual  terms. 


CHAPTER  VIII. 


APPLICATION  OF  THE  GENERAL  SUMMATION  FORMU- 
LAS TO  ARCHES  HAVING  A  HINGE  AT  EACH  SUP- 
PORT. 

WE  have  selected  an  arch  over  the  Douro  in  Portugal  to 
illustrate  the  application  of  the  summation  formulas,  as  the 
form  of  the  rib  is  such  that  none  of  the  common  formulas  can 
be  applied.  The  rib  is  crescent-shaped,  with  6  =  4.6  at  the 
crown  and  0.2  near  the  hinges. 

The  data  are  taken  from  Mhnoires  et  Compte  Rcndu  des  Tra- 
vaux  de  la  Society  des  Inge'nieurs  Civils  (Sept.  and  Oct.  1878), 
Memoir e  par  T.  Seyrig. 

The  superstructure  is  supported  symmetrically  at  points  A, 
B,  C,  D,  £,  etc.,  located  on  the  rib  as  shown  in  Fig.  54. 


-80;00  ----  '-  ---------------  •» 

All  dimensions  in  meters 

FIG.  54. 


The   linear   arch   used   in   the   computations  lies  midway 
between  the  flanges  of  the  arch  rib,  and  is  divided  into  twenty- 

182 


ARCHES  HAVING   A    HINGE  AT  EACH  SUPPORT.      183 


two  sections  as  indicated  in  Fig.  55.     The  coordinates  of  the 
points  of  division  are  given  in  the  following  table  of  data. 


FIG.  55- 


. 

2*. 

yls 

«*Aj 

A* 

1 

(I) 

(2) 

(3) 

(4) 

(5) 

X 

(6) 

9X 

(7) 

»M 
(8) 

*» 

(9) 

o 

I 

2.80 

3-00 

8.10 

0.246 

0.293 

276.58 

98.78 

296.34 

27.64 

•  a 

8.40 

9.00 

8.15 

0.588 

0.274 

1047.81 

124.74 

1122.66 

29.70 

3 

14.10 

14-55 

8.15 

I-  153 

0.264 

1450.04 

102.84 

1496.32 

30.87 

4 

20.40 

20.42 

8.80 

1.848 

0.253 

1983.49 

97-23 

1985.43 

34-73 

s 

25-25 

24.20 

3.80 

2.463 

0.242 

941.58 

37-33 

903.38 

15.70 

6 

31.00 

28.30 

9.80 

2.863 

0.236 

3002.97 

96.87 

2741.42 

4I-52 

7 

39-75 

32-75 

10.05 

3.486 

0.225 

3752.79 

94.41 

3091.92 

44-66 

8 

49-15 

36-85 

10.40 

3.758 

O.222 

5012.31 

101.98 

3757.96 

46.84 

9 

59-20 

40-35 

10.75 

4.220 

O.222 

6084.57 

102.78 

4147.17 

48.42 

10 

69.60 

42.25 

10.50 

4.609 

O.228 

6699.00 

96.25 

4066.56 

46.05 

ii 

80.00 

42.65 

5-25 

4.696 

O.228 

3814-40 

47-68 

2033.55 

23.02 

34066.54 

25642.71 

389-20 

In  the  above  table  columns  (i)  to  (5)  inclusive  contain  the  data  necessary  for 
the  determination  of  the  values  of  the  horizontal  thrusts  due  to  a  loading  of 
unity  at  any  point  being  considered.  Columns  (6)  to  (9)  inclusive  have  been 
computed  from  the  data  given. 

The  formula  to  be  applied  in  this  case  is  (242),  page  49,  or 


As 


*  Ax  sin 


As 


ll*Ax  sin  0  ) 

a  FT~ 'i 


/*  Ax  cos  0 


1 84 


A    TREATISE   ON  ARCHES. 


The  term  containing  Fx  in  the  numerator  is  very  small  and 
can  be  neglected  without  serious  error. 

Ax  cos  0  =  As  approximately.     Then  we  have 


As 


As 


TT     _ 

•**   j      — 


Since  the  first  term  of  the  numerator  and  the  entire  denom- 
inator are  constant  for  this  case,  we  may  place  them  at  once. 

Denominator  —  2(25642.71  -\-  389.20)  =  52064; 


~  =  34066.54. 


Therefore 


'/t  As 

34066.54  —  2(x  —  d)y  -j- 

H,  = ^ ^-P. 

52064 

There  remains  then  but  one  term  to  compute  as  the  position 
of  the  load  changes. 

Following  are  the  necessary  computations  for  determining 
the  values  of  Hl  for  loads  at  A,  JS,  C,  D,  and  E  respectively: 

Load  at  A. 
a  =  25.25,         k  —  0.157. 


Point. 

,-„ 

(*-.*£ 

5 

o 

0 

Since   a   must  be    less 
than  x,  where   the   term 

6 

7 
8 

5.75 
14.50 
23.90 

557-00 
1368.94 

2437-32 

(x  —  a)  is  employed,  we 
need  to  compute  only  the 

9 

33-95 

3489-38 

quantities    given   in    the 

10 

44-35 

4268.68 

table. 

ii 

54-75 

2610.48 

14731.80 

ARCHES  HAVING   A    HINGE  AT  EACH  SUPPORT.      185 

We  have,  substituting  the  several  values  given  above, 

H  _  ^34066.54  -  14731.80  __  19334.74 
52064  52064 


or 


or 


or 


Load  at  B. 
a  =  54.00. 


Point. 

x  —  a 

"-< 

9 

5-2 

534-45 

10 

15-6 

1501.50 

II 

26.O 

1239.68 

3275-63 

H  _  ^34066.54  -  3275-63  _ 

52064  52064 


H,  =  0.5914^. 


Load  at  C. 
a  =  64.4. 


Point. 

JT  —  a 

Ac 

(*  -  a}y  ^ 

10 

5-20 

500.50 

II 

15.60 

743-80 

I244-30 

_  34066-54  —  1244.30   32822.24 
52064      52064  " 

H,  =  0.6304^. 


186 


A    TREA  TISE   ON  ARCHES. 

Load  at  D. 

a  =  74.8. 

Point. 

x  -  a 

U-«),g 

IO 
II 

O 
5  JO 

o 
247-93 

247-93 

_  34066.54  -  247.93   33818.61 
-    52064      52064  r 


or 


H,  =  0.6495/1 

Load  at  E. 

a  =  80.00.    x  —  a  =  o  at  E.     Hence 
34066. 54  n 
H--^^P 
or 

H,  =  o.6543/>. 

Having  now  determined  the  values  of  Hl  for  each  load,  the 
stresses  in  the  arch  can  be  found  graphically  for  any  given  value 
of  P.  Since  the  arch  is  hinged  at  the  ends,  the  values  of  Fi 
will  be  the  same  as  for  a  straight  unconfined  beam. 

The  following  table  shows  the  values  of  Hl  obtained  above 
with  those  given  by  Seyrig: 


Load  at 

ff, 

(I) 

ff, 
(Seyrig) 
(2) 

Diff. 
(3) 

Formula  (91), 
page  29 

(4) 

Diff. 

(  i)  and  (4) 

(5) 

A 
B 
C 
D 
E 

o.37i3/> 
o.59i4/> 
o.6304/> 
0.6495/1 
O.6543/* 

o.37o/> 
o.592/> 
•0.63I/* 
o.bsoP 

—  O.OOI3/1 

—o.ooobP 
—  o.ooo6/> 
—  o.ooos/' 

o.35S/> 
0.653/1 

O-7I2/" 

0.742  p 

o  746/* 

O.OI33/' 
0.0616;° 
o.o8i6/> 
0.0925^ 

O.OQljP 

*  As  given  by  Seyrig  this  is  0.637;  but  as  he   gives  it  as  the  quotient  of 
2048.36  H-  3246.84,  which  is  0.6309,  it  is  evidently  a  typographical  error. 


ARCHES  HAVING  A    HINGE  AT  EACH  SUPPORT.      l8/ 


The  differences  in  column  (3)  are  very  small  and  unim- 
portant. 

To  show  the  error  in  applying  the  common  formula  to 
arches  where  the  moments  of  inertia  do  not  vary  according  to 
the  law  making  6  cos  <p  =  a  constant,  columns  (4)  and  (5)  have 
been  computed,  from  which  it  is  seen  that  an  error  of  about 
sixteen  per  cent  would  be  made  in  applying  formula  (91),  page 
29,  to  this  particular  arch. 

The  following  tables  give  the  loading  which  was  assumed 
in  designing  the  arch,  the  unit  being  1000  kilograms  : 
1°.  MOVING  LOAD  COVERING  THE  ENTIRE  ROADWAY. 


Load 
at 

A 
B 
C 
D 

Load 
in  ,000*. 

Coefficient 
of  .//,. 

», 

126.8 
62.4 
47.0 
40.5 

0.3713 
0-59M 
0.6304 
0.6495 

47  08 
36.90 
29.62 
26.30 

Since   the  arch  is  symmet- 
rically loaded,  the  total  value 
of 
H,  =  139-90  X  2 

=  279.80 

139.90 

II".  SYMMETRICAL  MOVING  LOAD  COVERING 
OF  THE  SPAN. 


METRES  IN  THE  CENTRE 


Load 
at 

Load 
in  1000*. 

Coefficient 
of//,. 

//,. 

A 
B 

18.6 

53-8 

0.3713 
0.5914 

6.90 

3I.8I 

Since  the  arch  is  symmet- 
rically loaded,  the  total  value 

C 

47.0 

0.6304 

29.62 

of 

D 

40.5 

0.6495 

26.30 

Hl  = 

94.63  X  2 

94-63 

= 

189.26 

III0.  NON-SYMMETRICAL  LOADING. 


Load 

at 

Load 
in  looo*. 

Coefficient 
of//,. 

HI- 

A 

II3.O 

0.3713 

41-95 

B 

58.0 

0.5914 

34-30 

C 
D 

48.0 
36.6 

o  .  6304 
0.6495 

30.25 
23.78 

D' 

3-1 

0.6495 

2.01 

C' 

o 

B' 

4.6 

0.5914 

2.72 

A' 

13-5 

0.3713 

4.91 

...   !3.9-92  = 

Total  /^     . 

A    TREATISE   ON  ARCHES. 


FIG.  56. 


ARCHES  HAVING   A    HINGE  AT  EACH  SUPPORT.      189 

The  stresses  in  the  various  pieces  composing  the  arch  can 
now  be  found  either  by  computation  or  by  graphics. 

The  stress  diagram  for  a  load  over  all  is  shown  in  Fig.  56. 

From  this  example  we  see  that  arches  having  a  variable 
moment  of  inertia,  not  following  the  law  #cos  <p  =  a  constant, 
can  be  treated  with  but  very  little  labor.  Most  of  the  compu- 
tations can  be  made  with  the  slide-rule,  and  furthermore  the 
computer  need  not  be  familiar  with  the  theory  at  all,  merely 
deducing  certain  quantities  mechanically,  these  quantities  to 
be  used  by  the  person  who  is  responsible  for  the  designing  of 
the  structure. 

The  effect  of  the  axial  stress  can  be  readily  seen,  since, 
.approximately,  it  occurs  only  in  the  denominator.  In  our 
example  the  axial  stress  term  =  778.4.  Then  the  denominator, 
neglecting  the  axial  stress,  amounts  to  52,064  —  778  =  51,286, 
and  the  results  obtained  by  using  this  denominator  correspond 
to  those  obtained  by  the  common  method  if  the  effect  of  the 
variable  6  could  be  considered. 

The  relative  error  made  in  omitting  the  axial  stress  term  is 
i^ff-3-  =  0.015,  or  1.5  per  cent,  an  error  of  no  practical  im- 
portance. 


CHAPTER  IX. 

APPLICATION  OF  THE  GENERAL  SUMMATION  FORMULAS 
TO  ARCHES  WITHOUT  HINGES. 

IN  order  to  illustrate  the  application  of  our  formulas  and 
to  show  to  what  degree  of  accuracy  they  lead,  we  will  compute 
the  values  of  //, ,  Ml ,  etc.,  for  a  parabolic  arch  having  mo- 
ments of  inertia  varying  according  to  the  law  0  cos  (f>  =  a  con- 
stant, by  the  summation  method  and  by  the  formulas  demon- 
strated in  Chapter  III. 

DATA. 

Span  =7=190;        Rise  =7=25; 

Load  =  a  concentration  P  or  Q  =  unity  at  points  designated. 

VERTICAL   LOADS.      (P  =  unity). 
(a)  Determination  of  Hl  by  Summation. 

10  o 


FIG.  57. 

Let  the  semi-arch  be  divided  into  ten  parts  as  shown  in  the 
figure,  and  the  quantities  shown  in  the  tables  determined. 


—  k\f   and    tan  0  =  yr (£/  — 


IQO 


ARCHES    WITHOUT  HINGES. 


The  moments  of  inertia  are  determined  for  the  section  at 
the  middle  points  of  As  or  points  having  the  abscissas  x.  Only 
the  relative  values  need  be  determined  now,  as  we  propose  to 
neglect  the  effect  of  the  axial  stress. 


DATA. 


Point. 

k 

- 

y 

AJ 

0 

A.? 

~o* 

Approximate 

I 

0.026 

5 

2-5 

II.  2 

.12 

IO.O 

26°   30' 

2 

.079 

15 

7-3 

IO.g 

.09 

23°   54' 

3 

.132 

25 

"•5 

10.7 

.07 

21°    12* 

4 

.184 

35 

15-0 

10.5 

•05 

18°  23' 

5 

.237 

45 

18.1 

10.3 

.03 

15°    29' 

6 

.289 

55 

20.5 

IO.2 

.02 

12°   30' 

7 

•  342 

65 

22.5 

IO.I 

.01 

9°  26' 

8 

•  395 

75 

23-9 

IO.  I 

1.  01 

6°  20' 

9 

•447 

85 

24-7 

IO.O 

.00 

3°  10' 

10 

.500 

95 

25.0 

5-o(i) 

.00 

5.0 

0°     0' 

95.0 

SUMMATION  TERMS. 


AJ 

A* 

AJ 

AJ 

Point. 

fa* 

''•T 

*6^ 

**£ 

I 

25 

62.5 

50 

125 

2 

73 

532-9 

150 

1095 

3 

"5 

1322.5 

250 

2875 

4 

ISO 

225O.O 

350 

5250 

5 

181 

3276.1 

450 

8i45 

6 

205 

4202.5 

550 

"275 

7 

225 

5062.5 

650 

14625 

8 

239 

57I2.I 

750 

17925 

9 

247 

610O.9 

850 

20995 

IO 

125 

3125.0 

475 

"875 

1585 

31647.0 

4525 

94185 

From  (221),  page  46,  remembering  that  the  terms  contain- 
ing Nx  and  Fx  are  to  be  omitted,  since  we  propose  to  neglect 
the  axial  stress,  we  have 


A    TREATISE   ON  ARCHES. 

VK'As 
"K'yAs      f     Bx    VyJs 

T~JT    ~PT  '  °* 

off. 


H,  = 


where  K'  =  Vjc  —  2P(x  —  a) ;  or  for  the  left  half  of  the  arch, 

K'  =  Px  -  P(x  -  Xa\ 
But  P  =  unity,  and  hence 

x>  a. 

K'  =  x  -  (x  -  a). 
Then 


and 


*  (  3) 


We  will  first  determine  the  constants  in  our  expression  for 
The  denominator  becomes 


95 


ARCHES    WITHOUT  HINGES. 


193 


and  we  have 


10406 

The  following  table  contains  the  quantities  to  be  substi- 
tuted in  this  equation 


PARTIAL  SUMS. 


(1) 

(2) 

(3) 

(4) 

a   9 

"I  • 

a  6 

°aT 

I 

5 

94060 

7800 

4475 

425 

2 

15 

92965 

22305 

4325 

1125 

3 

25 

90090 

34300 

4075 

1625 

4 

35 

84840 

42770 

3725 

1925 

5 

45 

76695 

46845 

3275 

2025 

6 

55 

65420 

45980 

2725 

1925 

7 

65 

50795 

39715 

2075 

1625 

8 

75 

32870 

27900 

1325 

1125 

9 

85 

II875 

10625 

475 

425 

10 

95 

O 

o 

In  this  table  the  summation  is  actually  taken  between  £/ 
and  (a  -\-  i),  for  when  x  =  a  the  combination  of  columns 
I  and  2  and  3  and  4  respectively  equal  zero. 

For  a  load  at  (i),  our  equation  gives  us 


H.  = 


94185  -(94060-7800)  -{4525— (4475— 425)116.68. 
10406 


Hl  =  O.O002.* 

In  like  manner  the  values  of  H,  can  be  found  for  all  the 
points  from  I  to  10  inclusive.  These  values  are  given  in  the 
annexed  table. 


*This  value  of  Hl  should  be  zero  when  all  mathematical  work  is  correct. 


194 


A    TREA  TISE   ON  ARCHES, 
VALUES  OF  Hl  FOR  A  LOAD  UNITY  AT — 


Point. 

i 

Hi 

I 

.026 

o.ooo 

2 

3 
4 

I 

7 
8 
9 

.079 
.132 
.184 
-237 
.289 
-342 
•  395 
•447 

O.I4T 

0-357 
0.640 
0.932 

.212 

•454 
.641 

•  757 

In    case    any    load    other 
than  unity  is  placed   at  any 
point,      the       corresponding 
value  of  //]  is  found  by  mul- 
tiplying the  load  by  the  cor- 
responding coefficient  HI  in 
this  table. 

10 

.500 

•797 

(b}  Determination  of  Hl  by  Integration. 
From  (91),  page  29,  we  have 


, 

4/1 

which  becomes  for  our  arch  with  load  unity 

H,  =  28.6^(1  -  £)'     or    H,  =  28.64, 

if  the  tables  are  employed. 

The  values  of  ffl  are  as  follows  : 

VALUES  OF  HI  FOR  A  LOAD  UNITY  AT— 


Point. 

4 

//, 

I 

.026 

.018 

2 

.079 

.152 

3 

.132 

•  377 

In    case    any    load    other 

4 

.184 

•643 

than  unity   is  placed  at  any 

5 

•  237 

•938 

point    the    value    of    H\    is 

6 

.289 

.201 

found    by    multiplying     the 

7 

•  342 

•447 

load    by    the    corresponding 

8 

•395 

•633 

value  of  Hi  in  this  table. 

9 

•  447 

•  744 

10 

•  500 

.787 

ARCHES    WITHOUT  HINGES. 


195 


(c)  Comparison  of  Results. 
VALUES  OF  H\  FOR  LOAD  UNITY  AT — 


Point. 

by  Summation. 

by  Integration. 

Difference. 

Relative  Diff. 
in  Per  Cent. 

j 

O.OOO 

.Ol8 

r 

2 

O.I4I 

.152 

+  .011 

«  1  •  •  • 
1  I  7 

3 

0-357 

•  377 

+  .O2O 

a*  1  3 

4 

0.640 

•  643 

+  .003 

•2.8   |  0.4 

5 

0.932 

•938 

+  .006 

Ir'  [0.6 

6 

.212 

.201 

—  .Oil 

Iri  fi-o 

7 

•454 

•  447 

-.007 

"&  1  0.5 

8 

.641 

•633 

-.008 

(frj  1  0.5 

9 

•  757 

•  744 

-.013 

§  |  0.8 

10 

•  797 

.787 

—  .OIO 

H  [0.5 

Let  a  load  of  one  ton  per  horizontal  foot  of  the  arch  be 
assumed,  and  determine  the  value  of  //",  for  a  load  over  all. 
Then,  by  summation, 

Hl  =  10(9.0325)2  =  180.65  tons; 

by  integration  (for  concentrations), 

//i  =  10(9.0475)2  =  180.95  tons; 

180.95  —  180.65  =  0.30; 
and  relative  error  equals 


We  will  now  determine  the  values  of  Ml  by  both  methods. 

(d)   DETERMINATION  OF  Aft. 
From  (225),  page  47, 

iKxAs^xAs        tKAs'SAs 


,*  _ 


in  which 


0       ft, 


196 


A    TREA  TISE   ON  ARCHES. 


The  following  table  contains  all  the  constants  entering  the 
above  equation.  Substituting  these  constants  and  the  values 
of  K,  we  obtain  an  equation  quite  simple  in  its  application. 


CONSTANTS. 


Point. 

JT 

y 

4 

>'t 

9 

0 

-4 

I 

5 

2.5 

25 

62.5 

50 

250 

125 

2 

15 

7-3 

73 

532-9 

150 

2250 

1095 

3 

25 

"•5 

"5 

1322.5 

25O 

6250 

2875 

4 

35 

15-0 

150 

2250.0 

350 

12250 

5250 

5 

45 

18.1 

181 

3276.1 

450 

20250 

8145 

6 

55 

20.5 

205 

4202.5 

55° 

30250 

11275 

7 

65 

22.5 

225 

5062.5 

650 

42250 

14625 

8 

75 

23  9 

239 

5712.1 

750 

56250 

17925 

9 

85 

24-7 

247 

6100.9 

850 

72250 

20995 

10 

95 

25.0 

250 

6250.0 

950 

90250 

23750 

9' 

105 

24.7 

247 

6100.9 

1050 

110250 

25935 

8' 

"5 

23-9 

239 

5712.1 

1150 

132250 

27485 

t 

125 

135 

22.5 
20.5 

225 
205 

5062.5 
4202.5 

1250 
1350 

156250 
182250 

28125 
27675 

5' 

145 

18.1 

181 

3276.1 

1450 

210250 

26245 

4' 

155 

15.0 

150 

2250.0 

1550 

240250 

23250 

3' 

165 

"•5 

1322.5 

1650 

272250 

18975 

2' 

175 

7-3 

73 

532.9 

1750 

306250 

12775 

I* 

185 

2-5 

25 

62.5 

1850 

3-12250 

4625 

3170 

63294. 

18050 

2.284750 

301150 

1585 

31647 

Determination  of  Constant  Factors. 

'As'x*As 
2-J-2—-—  =  190(2284750)  =  434,102,500; 

0   Ux    0       "x 

K  A  C\  * 

la  =  325,802,500. 


Hence 


Denominator  =  108,300,000; 
;—  =  18050     and     ^—jj—  —  2,284,750; 

18,050 

108,300,000 

2,284,750 


108,300,000 


=  O.2II. 


ARCHES    WITHOUT  HINGES. 


I97 


Therefore 


jurii'Jsr,^F  + 


'  a  A 

~  •*—^- 


As 


We  are  now  prepared  to  determine  the" value  of  Ml  for  a 
load  at  any  of  the  points  I  to  10  inclusive. 

The  substitutions  in  the  above  formula  are  quite  simple  as 
illustrated  by  the  detailed  deduction  of  Ml  for  a  load  at  point 
6  (see  table  of  Partial  Sums). 

PARTIAL  SUMS. 


Point. 

;   . 
Y 

a  ~o~ 

2**Jt 

a   0 

y*4* 

a   « 

ft 

t+ 

a   0 

I 

18000 

2284500 

301025 

1  80 

3M5 

2 

17850 

2282250 

299930 

170 

3072 

3 

17600 

2276000 

297055 

160 

2957 

4 

17250 

2263750 

291805 

150 

2807 

5 

16800 

2243500 

283660 

140 

2626 

6 

16250 

2213250 

272385 

130 

2421 

7 

15600 

2I7IOOO 

25776O 

1  20 

2196 

8 

14850 

2II4750 

239835 

no 

1957 

9 

14000 

2O425OO 

218840 

100 

I71O 

10 

13050 

1952250 

195090 

90 

1460 

9' 

12000 

I842OOO 

169155 

80 

1213 

8' 

10850 

1709750 

141670 

70 

974 

7' 

9600 

1553500 

"3545 

60 

749 

6' 

8250 

I37I250 

85870 

50 

544 

5' 

6800 

n6ioco 

59625 

40 

363 

4' 

52CO 

920750 

36375 

30 

213 

3' 

36OO 

648500 

17400 

20 

98 

2' 

1850 

342250 

4625 

10 

25 

l' 

0 

o 

o 

0 

0 

M.  = 


-1 


Load  at  Point  6. 

a  =  55- 
-  .oooi6f|i.2i(30ii50)  +  2213250  —  55(16250)} 

+  .0211  {l.2l(3I70)+  16250  -   55(130)} 

—  .000161(1684494)  =  —  280. 749  £ 
+  .0211(12942)          =+  273.076  )* 


i98 
Hence 


A    TREATISE    ON  ARCHES. 


M,  =  273.076  -  280.749  =  -  7-673- 
In   like   manner  the  values  of  M1  for  loads  at  any  other 
points  are  determined.     We  have  tabulated  below  the  values 
obtained  by  this  method. 


Load  at 

Value  of  M,  . 

Load  at 

Value  of  MI. 

I 

-     4-683 

tf 

4-  8.256 

2 

—    10.380 

8' 

+  9-490 

3 

—    12.923 

7' 

4-   9-603 

4 

—    12.641 

6' 

+   8.927 

5 

—    10.687 

5' 

+   7.363 

6 

—     7-603 

4 

+   5-410 

7 

-     3.887 

3' 

4-  3-037 

8 

—       O.2OO 

2' 

4-    I.  210 

9 

+      3.317 

l' 

4-  0.123 

10 

4     6.200 

The  corresponding  values  of  Mlt  as  obtained  from  Table 
VI,  are  as  follows : 


Load  at 

k 

^6 

MI  =  95J8, 
page  30. 

if  It 

by  Summation. 

out. 

I 

.026 

-.046 

-      4-370 

-      4-683 

+  0.313 

2 

.079 

-.108 

—    IO.26O 

—    10.380 

+   0.120 

3 

.132 

—  .133 

-    12.635 

—    12.023 

+    .288 

4 

.184 

-.132 

-    12.540 

—    12.641 

-j-   .101 

5 

•  237 

—  .112 

—    10.640 

—  10.687 

+    -047 

6 

.289 

-.O8l 

-      7.695 

-     7-603 

-    .032 

7 

•  342 

—.043 

-      4-085 

-     3-887 

-  .198 

8 

•395 

-.003 

—      0.285 

—      O.2OO 

-    .085 

9 

.447 

+.032 

+      3-040 

+      3.317 

+     -277 

10 

.500 

-J-.062 

+      5-890 

+      6.200 

+     -310 

9' 

•  553 

+  .084 

+      7.980 

+  8.256 

+     -276 

8' 

.605 

+  .096 

+     9.120 

+   9-490 

+     -370 

7' 

.658 

+•099 

+      9-405 

+   9.603 

4-    -198 

6' 

.711 

+•092 

+      8.740 

+   8.927 

4-     .187 

5' 

.763 

+.078 

+      7410 

+   7.363 

-     .047 

4' 

.816 

+  •057 

+      5-415 

+    5-410 

—     .005 

3' 

.868 

+•035 

+      3-325 

+     3-037 

-      .288 

2' 

.921 

+  •015 

+      1.425 

+      I.  210 

—     .215 

i' 

•974 

+.002 

+     0.190 

+      0.123 

—     -3*3 

—   62.510 

—    63.004 

+  61.940 

+   62.936 

-      0.570 

—      0.068 

ARCHES    WITHOUT  HINGES. 


199 


If  our  points  had  been  taken  closer  together,  the  positive 
and  negative  moments  would  have  been  practically  equal  for  a 
load  over  all. 

The  values  of  Vl  can  now  be  found  from  the  formula 


For  a  load  at  6, 

Fi  =  8.927  +  7-673 

The  values  of  y^  are  determined  from 


,    , 

(50),  page  17. 


«  /<?<7^/  at  6, 


10, 
6.200  +  0.500(95) 


the  correct  value  for  all  loads  being  in  this  particular  case 
|y  =  -§(25)  =  30,  showing  that  the  above  values  are  sufficiently 
exact  for  practical  purposes. 

I 


FIG.  58. 

Having  the  values  of  //",  ,  Vl ,  and  y^  determined,  the 
magnitudes  and  directions  of  the  resultants  can  be  found  as 
follows,  and  then  the  stresses  determined  by  the  usual  methods: 


2OO 


A    TREATISE   ON  ARCHES, 


Construct  the  centre  line  of  the  arch  to  any  scale.  For 
any  load  as  6,  make  EN  equal  the  corresponding  value  of  y0. 
Make  EF  =  Vl  ,  DF  =  H, ,  and  EG  =  P,  and  complete  the 
parallelogram  of  forces  :  then  ED  —  R^  and  DG  =  EO  =  R^ ; 
also,  AH=x^AK  =  y,,LC  =  y^  and  CM  =  xv 

HORIZONTAL   LOADS.      (Q  —  unity.) 

From  (232),  page  48,  introducing  the  constants  already 
found,  and  modifying  the  form,  we  have 

^K'As 


-, 6.68 


where 


10406 
K'  =  ( y  -  by>* 


10406 


0. 


From  the    tables  computed  for  vertical  loads  the  partia 
sums  required  above  are  readily  found. 


PARTIAL  SUMS. 


Point. 

vW  y*A* 

0       8x 

*£\1  yds 

^VJ» 

a    6x 

I 

31584-5 

1560 

85 

2 

31051.6 

1487 

75 

3 

29729.1 

1372 

65 

4 

27479.1 

1222 

55 

5 

24203.0 

IO4I 

45 

6 

20000.5 

836 

35 

7 

14938.0 

611 

25 

8 

9225.9 

372 

15 

9 

3125.0 

125 

5 

10 

o 

o 

0 

APPLICATION  OF   THE   GENERAL   SUMMATION.       2OI 


To  illustrate  the  application  of  the  formula  we  will  deter- 
mine the  value  of  //",  for  a  load  at  point  7. 
Load  at  7, 

b  —  22.5 


Then 


_ 

•CJi    - 


ff= 


/+  [14938  -22.5(61  1)]     \ 
_    \-  i6.68[6i  i  -22.5(25)1/ 

-  -  ~ 


10406 

-  16.68(756.6)] 


10406 
=  0.536. 


73' 


By  Table  XII,  H,  =  A,  =  0.537. 

The  following  table  contains  the  values  of  Hl  as  found  by 
two  methods  for  loads  at  points  I  to  10  inclusive. 


Point. 

£ 

HI  , 
by  Summation. 

by  Table  XII. 

Diff. 

I 

.026 

1.  000 

0.991 

.009 

2 

.079 

0.935 

0.930 

.005 

3 

.132 

0.839 

0.836 

.003 

4 

.184 

0.742 

0.740 

.OO2 

5 

•  237 

0.652 

0.651 

.OOI 

6 

.289 

0.585 

0.584 

.001 

7 

•  342 

0.336 

0-537 

.005 

8 

•395 

0.511 

0.511 

.002 

9 

•447 

0.501 

0.502 

.001 

10 

.500 

0.500 

0.500 

.000 

For  loads  on  the  right  of  the  point  10  we  have  merely  to 
subtract  the  value  of  77,  for  the  corresponding  load  on  the 
left  of  the  crown  from  unity. 

The  above  values  of  Hl  are  for  loads  acting  from  the  right 
towards  the  left,  and  hence  they  are  positive  and  the  same  in 
character  as  for  loads  acting  vertically  downward. 

For  bending-moments  Ml  we  have  from  (236),  page  48, 
introducing  the  constants  already  found, 


202  A    TREATISE    ON  AKCHES. 


' 

The  partial  sums  required  above  are  given  on  page  197. 

As  the  application  of  this  formula  is  precisely  the  same  in 
method  as  that  for  vertical  loads,  we  will  only  illustrate  its 
application  in  a  few  cases. 

Load  at  Point  10. 


: 


M    _        (    —  0.000162(281735)    =    —   46.956, 
)    +0.0211(2375)=    +    50113; 

or 

M,  =  +  3-I57- 
By  the  use  of  Table  XIII,  Ml  =  +  0.1250(25)  =  +  3.125. 

.    Load  at  Point  5  . 
b  =  18.1. 


Mt  = 


jfl    _        (    —    0.000l6|(2l6770)    =    —    36.128, 

{  +  0.021  1(1975)  =+  41.672. 

Hence 

W  =  +  5.544. 

By  the  use  of  Table  XIII,  Ml  =  +  5.450. 

The  above  results  indicate  a  close  agreement  in  the  two 
methods. 

To  determine  M3  it  is  necessary  to  merely  consider  a  as 
/-  *. 

The  method  of  procedure  is  now  parallel  with  that  outlined 
for  vertical  loads. 

Fig.  59  shows  graphically  the  results  obtained  by  the  two 
methods  somewhat  exaggerated. 


APPLICATION  OF  THE   GENERAL   SUMMATION.       2O$ 

The  close  agreement  of  the  curves  shows  clearly  that  the 
approximate  method  of  summation  is  quite  accurate  enough 
for  practical  purposes.  This  method  requires  considerable 


more  \vork,  but  it  has  the  advantage  of  being  approximately 
correct  for  any  form  of  arch  and  any  values  of  0,  the  circular  or 
elliptical  arch  requiring  no  more  labor  in  calculating  the  values 
of  Ht,  Mt,  etc.,  than  the  parabolic  arch. 


CHAPTER  X. 
THE  ST.  LOUIS  ARCH.* 

To  further  show  the  accuracy  of  the  results  obtained  by 
the  use  of  the  summation  formulas  we  will  compute  the  values 
of  //,  and  M}  for  the  well-known  St.  Louis  or  Eads  Bridge, 
using  the  data  given  in  the  History  *  of  the  bridge.  The  results 
given  by  Prof.  Woodward  were  computed  with  great  care  from 
formulas  deduced  ,to  fit  the  peculiarities  of  the  arch-rib. 

6  has  but  two  values  throughout  the  rib.  For  a  distance 
equal  to  one  twelfth  of  the  span  from  each  support  9  has  a 
constant  value,  and  between  these  two  sections  another  value 
which  is  uniform  throughout  that  section ;  thus  the  use  of  the 
formulas  of  Chapter  IV  is  prohibited. 

DATA. 

Span  =  /  =  519.2328  ft.     Rise  =/=  47-31  ft. 

Radius  =  R  =  736.0  ft.     00  =  20°  39'  i7//.92. 

Area  of  each  flange  for  -fa  the  span  at  the  ends  =  F  =  67 
sq.  in. 

Area  of  each  flange  in  centre  section  =  F  =  100.5  sq.  in. 

Depth  centre  to  centre  of  flanges  =  12  ft. 

Dead  load  =  I  ton  per  running  foot  horizontal. 

Live      "      =0.8"     " 

In  applying  our  formulas  the  linear  arch  will  be  assumed  to 
lie  midway  between  the  flanges  of  the  rib.  We  will  divide  this 
linear  arch  into  fifty-one  divisions,  as  shown  in  the  first  table. 
The  coordinates  x  and  y  will  be  computed  for  the  centre  points 

*  See  "A  History  of  the  St.  Louis  Bridge,"  by  C.  M.  Woodward  (St.  Louis, 
G.  J.  Jones  &  Co.,  1881). 

204 


THE   ST.    LOUIS  ARCH. 


205 


of  these  divisions,  and  the  moments  of  inertia  taken  at  the  same 
points. 

Since  the  areas  of  the  flanges  are  67  and  100.5  sq-  in-»  and 
the  distance  centre  to  centre  of  the  flanges  12  ft.  throughout, 
the  moments  of  inertia  will  be  in  the  ratio  of  two  to  three.  As 
we  propose  to  neglect  the  influence  of  the  axial  stress — as  was 
done  by  the  computers  for  the  structure  as  built — we  need  not 
concern  ourselves  about  the  actual  values  of  0,  but  use  relative 
values.  The  following  data  will  be  used  throughout  in  the 
computation  of  Hl  and  Ml : 

TABLE  OF  CO-ORDINATES,  ETC. 


Point. 

X 

y 

Ay 

4* 

4 

Relative  0. 

' 

6.6 

2-33 

4.83 

13.3 

14.14 

3 

2 

18.3 

6.52 

3.46 

IO.O 

10.58 

3 

3 

28.3 

9.98 

3-31 

10.53 

3 

4 

38.3 

13-22 

3.16 

10.48 

3 

5 

48.3 

16.31 

3.02 

10.44 

2 

6 

58.3 

19.18 

2.87 

10.40 

2 

7 

68.3 

21.98 

.65 

10.34 

2 

8 

78-3 

24-55 

•50 

10.31 

2 

9 

88.3 

27.06 

•43 

10.29 

2 

10 

98-3 

29-34 

.21 

10.24 

2 

ii 

108.3 

31-55 

.14 

IO.22 

2 

12 

118.3 

33-61 

.98 

10.19 

2 

13 

128.3 

35-45 

•77 

10.15 

2 

14 

138.3 

37-21 

.69 

IO.I4 

2 

15 

148.3 

38.76 

-55 

IO.I2 

2 

16 

158.3 

40.23 

.40 

IO.IO 

2 

17 

168.3 

41.56 

•25 

10.08 

2 

18 

178.3 

42-73 

.10 

10.06 

19 

188.3 

43-76 

.96 

10.04 

20 

198.3 

44-72 

.81 

10.03 

21 

208.3 

45-46 

0-74 

10.03 

22 

218.3 

46.  12 

0.51 

10.01 

23 

228.3 

46-63 

0.44 

10.01 

24 

238.3 

46.93 

0.40 

« 

10.007 

25 

251.4 

47-22 

0.12 

16.3 

16.30 

2 

. 

Determination  cf  //,. 

From  (220),  page  46,  remembering  that  the  terms  contain- 
ing Nx  and  Fx  are  to  be  omitted,  we  have  for  vertical  loads 


206 


A    TREATISE   ON  A  K  CUES. 


in  which  for  a  load  P  =  unity 


and 


~  *-  •  •  (223) 


We  note  that  only  the  quantities  enclosed  in  the  parentheses 
in  (222)  and  (223)  vary  with  a  change  in  the  location  of  the 
load.  We  will  first  compute  the  terms  which  are  constant. 

7|y^  =  156,868.7; 


2-j-  -=  125.0. 

Combining  these  quantities  and  multiplying  the  product  by 
2,  we  have  for  the  value  of  the  denominator  43324.1. 

^3     As 

~*y-7T  =  669403-3 


r-S-  =  16,863.4; 


THE   ST.    LOUIS  ARCH. 


207 


125.0 


_ 

3  * 


For  our  purpose  it  will  not  be  necessary  to  compute  //,  for 
a  load  at  each  point  of  division.  We  have  selected  points  2, 
4,  6,  10,  15,  20,  23,  and  25. 

The  following  tables  show  the  method  of  procedure  in  the 
determination  of  Hl  for  each  point  designated. 


FIRST  TERM  OF  NUMERATOR. 


First  Term  of 

Load  at 

j. 

Numerator: 

Point 

•^*yds 

//            A 

1  1        A 

I/  K'v  A 

No. 

o    e* 

£^—s 

a~£y— 

2—r~s 

a 

• 

0         * 

2 

669,403.3 

668,910.9 

74,606.5 

75,099-1 

4 

666,152.5 

153,035.0 

156,285.8 

6 

" 

656,225.8 

222,170.7 

235,348.2 

10 

" 

611,473.5 

322,5I4-9 

380,444.7 

15 

" 

495,454-3 

353.462.5 

527,411.6 

20 

" 

303,881.1 

260,155.3 

625,677.5 

23 

" 

152,699.9 

141,462.2 

658,165.5 

25 

" 

o 

O 

669,403.3 

SECOND  TERM  OF  NUMERATOR. 


Second  Term  of 

Load  at 

Numerator: 

Point 

ll  <iX  A* 

1  llX  A.S 

''/2 

I'l  Kf  A 

No. 

2~Q-~ 

2~iT~ 

a2~**x 

2~r~  <32'88) 

0      * 

a 

a 

0         * 

2 

16,863.4 

16,767.9 

2,136.6 

73,435-4 

4 

" 

16,534.9 

4.203.5 

149,  104.  i 

6 

' 

15,979-6 

5,791-1 

219,637.1 

10 

' 

14,265.3 

7,740.4 

340,139.0 

15 

' 

11,004.2 

7,909.3 

452,983-7 

20 

' 

6,519.9 

5,587.7 

524,124.8 

23 

1 

3,240.1 

3,002.8 

547,000.0 

25 

o 

o 

554,807.2 

208 


A    TREATISE   ON  ARCHES. 
VALUE  OF  Hi. 


Load  at  Point 
No. 

Numerator. 

Denominator. 

&\ 

2 

1,663.7 

43,324.1 

0.039 

4 

7,181.6 

0.165 

6 

I5-7II.I 

0.362 

10 

40,305.8 

0.930 

15 

74,427-9 

I.?-'? 

20 

101,542.7 

2-343 

23 
25 

111,163.5 
114,596.! 

2.565 
2.645 

If  now,  with  the  values  of  a  as  abscissas  and  the  correspond- 
ing values  of  Hr  as  ordinates,  points  be  located  on  sectioned 


2.6 

^o- 

2.4 

x< 

X^Z 

""" 

2.2 

A: 

/ 

2.0 

&. 

/ 

1.8      J" 

t 

7 

1.6      | 

/ 

1.4      ^ 

/ 

z 
< 

1.2      0 

/ 

A 

0) 

fe 

Q 

1.0     = 

/ 

a: 
H 

2 

0.8 

/ 

W 
0 

0.6 

/ 

0.4 

/ 

0.2 

/ 

! 

\ 

0.0 

^ 

x 

ii 

I 

II 

IT 

00 

20 

40 

60 

80 

100 
VAL 

120 

UES  C 

140 

>Fa 

160 

180 

200 

220 

240 

260 

FIG.  60. 

paper  and  a  smooth  curve  drawn  through  them,  the  value  of  //, 
for  any  value  of  a  can  be  readily  and  quite  accurately  deter- 
mined. 


THE   ST.    LOUIS  ARCH. 


209 


For  a  uniform  load  of  w  per  horizontal  unit  of  span  the 
total  value  of  //,  will  be  twice  the  area  included  between  the 
above  curve  (extending  from  the  support  to  the  crown)  and  the 
axis  of  abscissas  multiplied  by  w. 

Such  a  curve  is  shown  in  Fig.  60.  The  full  line  represents 
the  curve  located  by  the  above  values  of  H .  The  broken 
line  is  located  by  values  of  Hl  which  were  obtained  by  another 
computation  in  which  only  one  decimal  place  was  employed 
in  the  data. 

In  the  computations  for  the  St.  Louis  arch  uniform  loads 
were  assumed  as  follows  : 

For  dead  load  i.o  ton  per  lineal  foot. 
"     live       "     0.8    "      "        "          " 

In  the  history  of  the  bridge  the  values  of  H^  are  given  for 
a  load  extending  from  the  support  up  to  each  of  eight  points 
of  division.  The  corresponding  points  are  marked  I,  II,  III, 
etc.,  in  Fig.  60. 

The  following  table  shows  the  relation  between  the  values 
of  HI  given  in  the  history  of  the  bridge  and  those  obtained 
from  Fig.  60. 

MOVING  LOAD  OF  0.8  TON  PER  LINEAL  FOOT.     VALUES  OF  H\. 


Load  up  to 

History. 

Fig.  60. 

Difference. 

Remarks. 

I 

Tons. 
8.IO 

Tons. 
8.04 

O.O6 

The  values  in  the 

II 

56.20 

55-86 

0-34 

third  column  were 

III 

155-20 

154.78 

0.42 

obtained  from  Fig. 

IV 

286.60 

286.56 

0.04 

60  by  taking  T8ff  the 

V 

418.10 

413.34 

0.24 

area    between   the 

VI 

517.00 

517-26 

0.26 

full    line   and   the 

VII 

565.10 

565.08 

0.02 

axis  of  a. 

over  all 

573-30 

573-12 

o.i  8 

1 

The  above  table  shows  almost  perfect  agreement  between 
the  exact  and  approximate  methods.  The  errors  are  of  no 
practical  importance.  They  exist  only  in  the  decimal  figures, 
which  are  quite  likely  to  be  in  error  by  either  method. 

For  a  load  over  all  with  w  =  O.8  ton  the  area  between  the 


2IO  A    TREATISE   ON  ARCHED. 

broken  line  and  the  axis  of  a  is  559-3,  being  in  error  14  tons, 
or  a  little  over  2  per  cent.  Even  this  is  of  no  practical 
importance. 

'  We  will  now  show  that  the  effect  of  the  axial  stress,  which 
was   neglected    in    the   calculations   made    for   the   St.   Louis 
Bridge,  is  very  much  larger  than  any  error  which  is  likely  to 
be  made  by  using  the  summation  formula. 
In  (221)  we  have  in  the  numerator 

A*  sin  0 


o     XFX        o       Fx 

Since  we  used  only  relative  values  for  B,  it  will  be  necessary 
to  introduce  a  factor  in  the  above  expression.     For  the  area 

67,     6  =  2—  .  ( — )    =2(16.75)     approximately.      Therefore 
144    \  2  / 

2      2       =  2(8.37)  =  the  factor  required  ;  then 

7/a      Ax          a         Ax  sin  d> 
rxjr  =  2^8.37       g        • 


This  is  very  small  in  comparison  with  the  remaining  terms 
in  the  numerator,  and  hence  can  be  neglected  without  serious 
error. 

In  the  denominator  of  the  same  equation  we  have 

l!*Ax  cos  d>              '/»        Ax  cos  0 
+  2      p^  or     2^8.37 ^-r- 

We  may  replace  Ax  cos  0  by  As  nearly,  and  have 

v*As 
16.7 '4^-p-  =  8300,  about. 

Then  the  denominator,  when  the  effect  of  the  axial  stress  is 
considered,  becomes 

43,324+2(8300)  =  59924. 


THE  ST.  LOUIS  ARCH.  211 

=  1.38,  or  the  values  of  Hl  obtained  above  are  too 

large,  and   should   be   divided   by  1.38   to   obtain  the  values 
which  include  the  effect  of  the  axial  stress  (see  page  283). 


Deduction  o 
Neglecting  the  axial  stress  term,  (225),  page  47,  becomes, 


where 


The  values  of  the  constant  terms  are  as  follows : 

1  As  l  x*As 

2  -  =  250         ^-^—  =  22,035,617. 

0  Vx  0        Vx 

^  =  64,887. 

0     "* 

Hence  the  denominator  =  1,298,530,726. 

*  ^  ^  ^ 

—  =  0.0169696; 


D 

-2— —  =  0.000050046. 
o  V* 

Then  our  equation  becomes 

M,  =  0.000050046^-^^  —  0.0169696^-^—. 

The  following  tables  contain  the  necessary  quantities  for 
substitution  in  the  above  equation,  using  the  values  of  Hl 
found  above  and  the  same  points  for  the  location  of  the  loads. 


212 


A    TREA  T1SE   ON  ARCHES. 
FIRST  TERM. 


Load  at 
Point 
No. 

^'f^~ 

JgA 
*  «* 

t.*4* 

a^ 

tjEfJt 

o       * 

First  Term. 

2 

83,226.4 

22,034,233.5 

1,185,691.6 

20,931,768 

1047.551 

4 

352,111-5 

22,026,302.9 

2,472,605.5 

19,905,809 

996-205 

6 

772,5ii-3 

21,996,451.0 

3,731,410.1 

19.037,552 

952.750 

10 

1,985,268.8 

21,851,139.9 

6,123,034.1 

17,713,374 

886.483 

15 

3,664,093.9 

21,427,952.6 

8,754,026.2 

16,338,020 

817.652 

20 

4,999,983-7 

20,623,541.6 

10,816,260.6 

14,807,264 

741.044 

23 

5,473,733-7 

19,906,568.2 

11,703,816.8 

13,676,485 

684.453 

25     ' 

5,644,454.5 

19,107,367.4 

12,073,221.3 

12,678,600 

634.513 

25' 

5,644,454.5 

18,522,875.2 

12,276,321.8 

11,891,007 

595.096 

23' 

5,473.733-7 

17,704,577.3 

12,502,906.4 

10,675,404 

534.26I 

20' 

4,999,983.7 

16,250,248.4 

12,292,268.3 

8,957.963 

448.310 

15' 

3,664,093.9 

13,141,610.2 

10,927,427-6 

5.878,276 

294.184 

10' 

1,985,268.8 

9,041,875.9 

8,102,179.8 

2,924,964 

146.382 

6' 

772,5"-3 

4,928,737.8 

4,623,750.2 

1,077,498 

53-924 

4' 

352,in.5 

2,964,105.3 

2,835,176.7 

481,040 

24.074 

2' 

83,226.4 

1,237,593.8 

1,209,345.9 

"1,514 

5.580 

SECOND  TERM. 


Load  at 
Point 
No. 

*&£ 

o  o* 

& 

'& 

IK*, 

0°* 

Second 
Term 

2 

320.6 

64,791.9 

4,423.8 

60,688 

1029.864 

4 

1,356.5 

64,558.9 

8,990.4 

56,925 

965.994 

6 

2,976.2 

64,003.6 

13,077.6 

53,902 

914.697 

10 

7,648.5 

62.289.3 

20,026.3 

49,9" 

846.978 

15 

14.116.4 

59,029.2 

26,444.2 

46,701 

792.502 

20 

19,263.1 

54,544-9 

30,371.8 

43,436 

737-094 

23 

21,088.3 

51,265.1 

31,536.4 

40,816 

692.646 

25 

21,726.0 

48,023.9 

31,420.7 

38,329 

650.431 

25' 

21,726.0 

45,841.4 

31,287.9 

36,279 

615.648 

23' 

21,088.3 

42,980.1 

31,075.4 

32,993 

559-8/7 

20' 

19,263.1 

38,305-6 

29,455.4 

28,113 

477.069 

15' 

14,116.4 

29,461.9 

24,698.2 

18,880 

320.386 

10' 

7.648.5 

19,249.6 

»7.307.4 

9-590 

162.750 

6' 

2,976.2 

10,032.0 

9,425.4 

3,582 

60.798 

4' 

1.356.5 

5,895.6 

5,645.8 

1,  606 

27-257 

2' 

320.6 

2,414.3 

2,359-2 

375 

6-375 

In  making  the  computations  above  three  decimal  places 
were  used  throughout.  These  have  not  been  given,  hence  the 
last  figures  may  not  exactly  check 


THE   ST.    LOUIS  ARCH. 

ALUES   OF  MI. 


213 


Load  at 

Computed  Values 

Load  at 

Computed  Values 

Point  No. 

of  M*  . 
Load  Unity. 

Point  No. 

ot  Aft 
Load  Unity. 

2 

-  17-7 

25' 

+  20  6 

4 

-  30-2 

23' 

+  25.6 

6 

-38.1 

20' 

+  28.8 

IO 

-  39-5 

15' 

+  26.2 

15 

-  25.2 

10' 

-4-  16.  4 

20 

—    4.0 

6' 

+    6-9 

23 

+    .8.2 

4' 

+    3-2 

25 

+  15-9 

2' 

+    0.8 

With  the  values  of  a  as  abscissas  and  those  of  Ml  as  ordi- 
nates  the  curve  shown  in  Fig.  61,  page  215,  can  be  located. 
The  following  table  shows  the  agreement  between  the  values 
given  in  the  History  and  those  obtained  from  Fig.  61. 

COMPARISON  OF  VALUES  OF  M\.     w  =  0.8  TON. 


Load 

Upto- 

Values  Given 
in  History 
of  Bridge. 

Values  from 
Fig.  61. 

Difference. 

Percentage  of 
Computed 
Values. 

(*) 

'(2) 

.(3) 

(4) 

I 

—  1206 

-  1244 

+    38 

3-0 

11 

-  3"4 

-  3224 

4-  no 

3-4 

III 

-  4034 

—  -  4226 

4-  192 

4-5 

IV 

-3588 

-  3848 

-f  260 

6-7 

V 

-  2235 

—  2529 

+  294 

It.  6 

VI 

-     782 

—  1123 

-f  341 

30.3 

VII 

+      70 

—     282 

+  352 

124.8 

VIII 

+    232 

-     128 

+  36o 

281.2 

COMPARISON  OF  VALUES  OF  Mt. 


I 

-f  161 

-f  154 

-   7 

4-5 

II 

-f  1013 

+  985 

-  28 

2.8 

III 

-f-2466 

-f-  2401 

-  65 

2.7 

IV 

+  3821 

+  3720 

—  101 

2.7 

V 

+  4266 

+  4008 

-258 

6.3 

VI 

-f  3346 

+  3096 

—  250 

8.0 

VII 

+  M38 

+  1116 

-  322 

28.7 

VIII 

+  232 

-  128 

+  360 

281.2 

214  A    TREATISE   ON  ARCHES, 

In  column  (3)  the  positive  sign  indicates  that  the  values  in 
column  (2)  are  too  large. 

Here  we  see  that  the  agreement  in  values  is  not  as  close  as 
in  the  values  of  //",,  but  we  also  note  that  the  greatest  discrep- 
ancies occur  in  the  small  and  non-important  values. 

The  maximum  negative  value  of  Mt  is  in  error  —  but  4.5  per 
cent  and  the  maximum  positive  value  6.3  per  cent  —  errors  which 
are  of  little  importance. 

The  negative  area  in  Fig.  61  is  about  4.5  per  cent  too  large 
and  the  positive  about  6.3  per  cent  too  small.  Now  since 
in  this  particular  case  the  difference  between  these  areas  is 
small,  we  readily  see  why  our  discrepancy  for  a  load  over  all 
is  so  large.  The  heavy  broken  line  in  Fig.  61  represents  the 
correct  curve. 

For  practical  purposes  our  curve  is  quite  exact,  and  will 
give  results  as  near  the  truth  as  any  of  the  common  methods 
in  their  special  cases. 

Of  course  the  particular  advantage  in  the  summation 
method  is  its  adaptability  to  any  case  of  the  symmetrical  arch. 

Temperature. 

Data.—et°   =  0.000527,  where  t°   =  80°,  /  =  520,  E   = 
1944000  tons  per  square  foot. 
Value  of  Ht.  —  From  (239), 

„  _  Eefl  _  532734 
~~       — 


In  this  case  the  actual  values  of  dx  must  be  employed  in  the 
denominator,  or 


THE   ST.    LOUIS  ARCH. 


215 


„§[ 

5 

rf 

£ 

1 

7 

1 

1 

P 

s 

7 

1 

/ 

i 

=/ 

— 

-~l 

f 

1 

i 

> 





-1 

\ 

1 

\ 

1 

- 

\ 

1 

H3I 

N30 

V1 

1 

§ 

\ 

g 

\ 

§ 

N 

* 

§ 

8 

\ 

8 

\ 

g 

/ 

=_ 

3 

§ 

y 

Q 

^x 

^ 

s?TBlN  o 

2 

3AI±IS(Dd 

2l6  A    TREATISE   ON  A  ACHES. 

From  the  history  of  the  bridge, 

Ht  =  204.9. 

205.9—204.9  =  i,  or  an  error  of  about  one  half  of  I  per  cent. 
Value  o/M,.  —  From  (240),  page  49,  we  see  that 


M,  =#;  VT  =  205.9(32-88)  =  6769.9. 
i|f 

o    Vx 
From  the  history  of  the  bridge, 

M,  =  6747. 
6769.9  —  6747  =  22.9,  or  an  error  of  about  3.4  per  cent. 


CHAPTER  XI. 
THE  SPANDREL-BRACED  ARCH. 

THE  so-called  spandrel-braced  arch  usually  consists  of  an 
arched  bottom  chord  and  a  horizontal  top  chord  connected  by 
a  system  of  web-bracing.  Evidently  the  formulas  of  Chapters 
III  and  IV  cannot  be  applied  even  approximately  to  this 
form  of  arch.  The  summation  formulas,  however,  enable  us 
to  consider  this  type  of  arch  either  with  or  without  hinges  with 
comparatively  little  more  labor  than  required  for  the  ordinary 
form  having  a  variable  6. 

To  illustrate  the  method  to  be  pursued  we  will  take  the 
case  of  a  proposed  design  for  a  bridge  over  the  river  Douro 
by  Mr.  Max  Am  Ende  and  Messrs.  Handyside  &  Co.* 

The  form  and  general  dimensions  of  the  bridge  are  given 
in  Fig.  62. 

c p 

ffi 


All  dimensions  in  meters 
.86,? i. 


FIG.  62. 


When  the  general  form  of  the  structure  has  been  decided 
upon,  the  first  step  is  to  approximately  determine  the  sections 

*  Design  for  a  bridge  over   the  river   Douro  by  Mr.  Max  Am  Ende   and 
Messrs.   Handyside  &  Co.;  Engineering,  London,  1881. 

21 


218 


A    TREATISE   ON  ARCHES. 


of  the  various  members  by  the  formulas  of  Chapter  III  or  IV, 
using  for  the  linear  arch  the  parabola  or  circle  which  lies 
approximately  midway  between  the  two  chords. 

For  the  application  of  the  summation  formulas  the  linear 
arch  is  assumed  to  pass  through  the  centres  of  gravity  of  each 
vertical  section.  (Of  course  in  both  cases  mentioned  above  the 
linear  arch  must  pass  through  the  supports.)  The  method  of 
procedure  is  now  the  same  as  already  explained  for  the  arch 
with  a  hinge  at  each  support  and  the  arch  without  hinges. 

In  computing  the  values  of  6  for  each  section  the  moments 
of  inertia  of  the  flange  sections  about  an  axis  passing  through 
their  centres  of  gravity  may  be  neglected  and  the  moment  of 

each  flange  be  taken  as ,  where  h  is  the  distance  centre  to 

4 
centre  of  the  flanges. 


Douro  Spandrel-braced  Arch, 

Let  ABODE,  Fig.  62,  represent  one  half  of  the  bridge,  and 
suppose  the  approximate  dimensions  of  members  and  the  linear 
arch  have  been  determined.  We  will  divide  the  linear  arch 


FIG.  63. 

into  twenty  equal  parts,  measure  the  co-ordinates  at  the  centre 
•of  each  division,  and  take  the  moments  of  inertia  at  the  same 
points. 

Following  are  the  data  required  for  the  determination  of 


THE  SPANDREL-BRACED    ARCH. 


219 


DATA. 


Divi- 
sion. 

*AJ 

*Ajr 

*  by 

X 

y 

*sin<fr 

*cos* 

*    6x 

1000 

V* 

I 

10  2 

7.0 

7-4 

3-5 

3-7 

0-735 

0.68 

20.73 

5.60 

2 

7.0 

7-3 

10.5 

ii.  i 

0.720 

0.70 

11.23 

4.70 

3 

7.6 

7-1 

17.8 

18.2 

0.685 

0-73 

21.64 

5-00 

4 

7-8 

6.6 

25-5 

25-1 

0.630 

0.77 

40.41 

4.80 

5 

8.2 

6.2 

33-5 

31-5 

O.6oo 

0.80 

87.07 

4-80 

6 

9.0 

4.8 

42.1 

37-0 

0.480 

0.88 

157-57 

4.36 

7 

9-5 

3-9 

51-4 

41.4 

0.367 

o-93 

101.91 

4-30 

8 

IO.O 

2-3 

61.2 

44-4 

0.235 

0.97 

51.96 

4-18 

9 

IO.I 

1.2 

71.2 

46.2 

0.122 

0-99 

29.27 

4.00 

10 

IO.2 

0.4 

81.2 

47.0 

0.045 

I.OO 

17.84 

4.04 

DATA. 


*  1000 

1000 

1000 

1000 

1000 

*IOOO 

Divi- 

AJ 

1A* 

Aj 

I,*** 

XV** 

A.t  co$ 

sion. 

•"» 

J<rx 

0X 

'I* 

6X 

?* 

I 

0.492 

1.820 

1.722 

6-734 

6.371 

2.525 

2 

0.892 

9.901 

9.366 

109.901 

103.962 

2-499 

3 

0.471 

8.572 

8.384 

156.010 

152.589 

2.320 

4 

0.252 

6.32; 

6.426 

156.757 

161.292 

2-055 

5 

0.117 

3-68^ 

3-9I9 

116.077 

123.448 

1.817 

6 

0.065 

2.405 

2.736 

88.985 

101.232 

1-367 

7 

O.IOO 

4.140 

5.140 

171.396 

212.796 

1.251 

8 

o.  196 

8.702 

11-995 

386.368 

532.578 

1.109 

9 

0.348 

16.077 

24-777 

742-757 

II49.3I7 

1.043 

10 

0.572 

26.884 

46.446 

1263.548 

2182.962 

0.850 

3-505 

88.511 

I20.9II 

3198.533 

4726.547 

16.836 

From  (221),  page  46,  we  have,  neglecting  the  axial  stress 
term  in  the  numerator, 


'*K'yAs      7    0, 


#,= 


7*  cos  0 


*  Data  given  by  Mr.  Max  Am  Ende. 


22O  A    7^REATISE   ON  ARCHES. 

or 

H,  = 


1961,  say  2000 
where 

SQ*_Su$-« 


and 

'J* 


for  /*  =  unity. 

For  a  load  at  10,  a  —  81.2.       Then 

-  {o}  -  4726.5; 


For  a  load  at  9,  ^  =  71.2.     Then 

i^L>_£  =  47265  _  |2I83  _  7I.2(26.9)|  =4458.7 
and 


^-=121  —  {46.4  -  71.2(0.572)}  =  115.2. 


.  4458.7-115.^25.25) 

2000  //5> 

In   like    manner  the  values  of  Hl  for  loads  at  the  other 
points  are  obtained.      The  following  table  contains  the  values 


THE  SPANDREL-BRACED   ARCH. 


221 


of .//,  for  each  division,  the  interpolated  values  for  the  ends  of 
the  divisions,  and  the  values  given  by  Mr.  Max  Am  Ende,  who 
took  his  origin  of  co-ordinates  at  the  crown  and  measured  the 
xs  and  /s  to  the  extremities  of  the  divisions.  He  then  sub- 
stituted the  proper  quantities  in  *  three  equations,  which  he 
demonstrates,  and  eliminated  all  unknowns  but  Ht. 

COMPARISON  OF  THE  VALUES  OF  Hi 


Divis;on  No. 

#i 

ffi  at  End  of 
Divisions, 
Fig.  64. 

«ii 

Max  Am  Ende. 

I 



0.018 

0.032 

2 

0.037 

0.060 

0.109 

3 

0.117 

0.167 

O.2O4 

4 

0.218 

0.274 

0.302 

5 

0.323 

0.385 

0.402 

6 

0.440 

0.500 

0.509 

7 

0.550 

0.617 

0616 

8 

0.673 

0.725 

0.701 

9 

0.775 

O.SlO 

0.762 

10 

0.835 

0.840 

0.792  (?) 

0.8 

^~ 

0,7 

/ 

~£ 

•^~ 

0.6 

/ 

il'J 

0.5  ^ 

/ 

0.4? 

£ 

f 

0.3> 

j£ 

V 

0.2 

A 

^ 

0.1 

4! 

0 

' 

.»*** 

^ 

r 
3 

LOAD 
4 

AT 

"  5 

=>OIN' 
6 

r  NO. 

7 

8 

9 

10 

FIG.  64. 


*  As  far  as  known  by  the  author,  Mr.  Max  Am  Ende  was  the  first  to  suc- 
cessfully treat  the  fixed  arch  with  variable  9,  using  the  summation  formulas  By 
some  manipulation  his  three  formulas  can  be  reduced  to  our  general  forms. 


222  A    TREATISE   ON  ARCHES. 

We  see  from  Fig.  64  that  our  values  lie  above  and  below 
those  given  by  Mr.  Max  Am  Ende,  and  that  the  areas  between 
the  curves  located  by  both  series  of  values  and  the  axis  of  a 
are  very  nearly  equal,  that  is,  for  a  uniform  load  over  all  the 
values  of  Hl  would  be  practically  equal.  We  could  not  expect 
any  closer  agreement  in  values,  considering  the  difference  in 
method  and  the  very  approximate  values  of  x  and  y  which  we 
used 

It  will  not  be  necessary  to  take  up  the  deduction  of  J/,, 
V^  etc.,  as  the  method  of  procedure  is  precisely  the  same  as 
that  employed  for  the  St.  Louis  arch. 

The  more  common  form  of  the  spandrel-braced  arch  is 
hinged  at  each  support.  The  method  of  treatment  is  prac- 
tically the  same  as  outlined  above ;  only  the  formulas  for  the 
hinged  arch  are,  of  course,  used. 


CHAPTER  XII. 
THE  MASONRY  ARCH. 

UNDER  this  heading  we  will  include  arches  constructed  of 
stone,  brick,  and  concrete  having  spans  of  at  least  twenty-five 
feet. 

Before  considering  the  many  types  of  masonry  arches  we 
will  first  consider  a  type  which  is  amenable  for  calculation 
by  the  formulas  deduced  for  the  elastic  arch.  This  type 
consists  of  an  arch-rib  of  masonry,  with  joints  carefully  made 
and  as  thin  as  practicable.  At  regular  intervals  this  arch 
supports  thin  lateral  walls,  which  in  turn  carry  small  arches  or 
slabs  which  support  the  roadway.  At  the  abutments  the  arch 
is  protected  from  any  horizontal  pressures  by  retaining  walls. 
The  general  features  of  this  type  are  shown  in  Fig.  65.* 


FIG.  65. 

The  dead  weight  of  this  style  of  bridge  consists  (i°)  of  the 
weight  of  the  masonry  in  the  rib  proper,  and  (2°)  the  weight  of 

*  See   "Bericht  des  Gewolbe-Ausschusses.     Sonderabdruck  aus  der  Zeit- 
schrift  des  Osterr.  Ingenieur-  und  Architekten-Vereines,"  No.  20-34,  1895. 

223 


224  A    TREA  TISE   ON  ARCHES. 

the  material  above  the  arch  which  is  transmitted  to  the  rib 
through  the  thin  lateral  walls.  The  forces  acting  upon  the 
arch-ring  are  evidently  vertical. 

Now  since  any  rectangular  masonry  joint  will  have  the 
same  kind  of  stress  at  all  points  when  the  resultant  pressure 
upon  the  joint  is  applied  within  the  middle  third,  our  arch-ring 
will  be  in  compression  throughout  if  the  equilibrium  polygon 
lies  within  the  middle  third  of  each  section.  Then  if  the  effect 
of  the  mortar  joints  be  neglected  the  masonry  rib  will  behave 
quite  similarly  to  an  elastic  rib,  and  hence  we  may  consider 
the  formulas  already  demonstrated  as  applicable  in  this  case. 

If  the  skew-backs  are  well  fitted  and  the  abutments  or 
piers  supporting  the  arch  practically  immovable,  then  the 
masonry  rib  is  fixed at  the  ends,  or  at  least  more  nearly  fixed 
than  hinged,  as  long  as  the  equilibrium  polygon  remains  within 
the  middle  third  of  the  section. 

Since  the  arch-ring  is  necessarily  made  up  of  many  pieces 
where  either  stone  or  brick  is  employed,  it  is  practically 
impossible  to  so  construct  the  arch-ring  that  there  will  not  be 
more  or  less  change  in  the  position  of  the  axis  when  the  false- 
works or  centring  is  removed.  As  a  consequence  the  true 
position  of  the  equilibrium  polygon  in  the  arch  as  constructed 
is  somewhat  uncertain. 

To  avoid  this  uncertainty  in  the  location  of  the  equilibrium 
polygon,  it  is  advisable  to  place  in  three  or  more  joints  which 
divide  the  ring  symmetrically  some  material,  as  lead,  covering 
the  middle  third  of  the  joint.  This  locates  the  polygon  within 
the  limits  of  the  area  of  the  lead  plates,  and  hence  the  maximum 
possible  thrusts  at  these  joints  can  be  determined. 

After  the  falseworks  are  removed  and  the  arch  with  its 
spandrels,  etc.,  completed,  these  joints  can  be  filled  with 
cement,  and  become  fixed  at  the  ends  for  any  additional  loads.* 
This  method  is  successfully  followed  by  German  engineers. 

For  arches  of  the  above  type  all  loads  are  considered  vertical, 
aud  the  arch-rib  is  assumed  to  be  without  hinges  for  moving  loads. 

*  See  page  229. 


THE  MASONRY  ARCH.  22$ 

As  in  all  arch  designs,  the  general  dimensions  must  be* 
assumed,  and  then  the  corresponding  loads  computed  and  the 
equilibrium  polygons  f  drawn  to  determine  if  they  lie  within 
the  middle  third  of  the  arch-ring  assumed,  and  further,  to  be 
sure  that  the  intensity  of  the  pressure  at  any  point  in  the  rib 
does  not  exceed  the  safe  strength  of  the  material  and  that 
frictional  stability  is  not  exceeded. 

Having  decided  upon  the  shape  of  the  arch,  the  span  and 
rise  of  the  axis  being  assumed,  the  next  dimensions  required 
are  the  thickness  of  the  rib  at  the  crown  and  that  at  the  skew- 
backs.  The  assumption  of  these  dimensions  can  be  made  with 
the  aid  of  Table  XXX. 


THICKNESS   OF   ARCH-RING  AT  THE   SKEW-BACK. 

Theory  (except  in  hinged  arches),  practice,  and  appearances 
demand  that  the  depth  of  the  arch-ring  at  the  skew-back 
should  be  somewhat  greater  than  at  the  crown.  For  vertical 
forces  the  horizontal  thrust  is  constant  throughout  the  arch, 
and  hence  the  axial  thrust  increases  as  the  secant  of  the  angle 
of  inclination  of  the  axis.  The  thickness  of  the  rib,  however, 
should  increase  more  rapidly  than  the  secant  of  this  angle, 
since  it  is  seldom  that  the  equilibrium  polygons  follow  the 
centre  of  the  arch-ring.  As  the  polygon  departs  from  the 
centre  of  the  ring  the  maximum  intensity  of  the  pressure  upon 
the  joint  changes  quite  rapidly,  being  twice  the  average  intensity 
when  the  polygon  passes  through  the  third  point  of  the  joint. 

Having  decided  upon  the  depths  of  the  crown  and  the 
skew-backs,  the  arch-ring  can  be  drawn  to  scale. 

*  See  Alexander  and  Thomson's  direct  method  for  proportioning  masonry 
arches,  page  234. 

\  The  graphic  method  is  preferred  for  the  preliminary  investigations,  being 
much  shorter  than  the  algebraic  methods,  and  quite  accurate  enough. 


226  A    TREATISE   ON  ARCHES. 


EQUILIBRIUM    POLYGON   FOLLOWING  THE   AXIS    OF  THE 
ARCH-RING. 

The  ideal  arch  would  be  one  in  which  the  pressure  over 
the  area  of  each  joint  is  uniform,  or  the  resultant  pressure 
would  pass  through  the  centre  of  each  joint  of  the  arch-ring. 
This,  of  course,  is  impossible  when  the  loading  is  movable  ;  but 
for  the  dead  load  of  the  structure  the  various  parts  can  be  so 
located  that  the  equilibrium  polygon  will  very  nearly  pass 
along  the  axis  of  the  arch-ring. 

Now  since  the  dead  load  is  usually  much  greater  than  the 
live  load,  if  the  arch  be  designed  so  that  the  equilibrium  polygon 
follows  the  axis  for  the  dead  load  and  a  live  load  over  all,  the 
ring  will  be  safe  usually  for  a  variable  moving  load. 

The  loading  necessary  to  make  the  equilibrium  polygon 
follow  the  axis  can  be  obtained  approximately  as  follows : 

Assume  the  dimensions  of  the  arch-ring  and  draw  it  to 
scale  as  shown  in  Fig.  66.  Determine  the  distance  mp  and 
the  location  of  the  points  a,  b,  c,  etc.,  where  the  lateral  walls 
rest  upon  the  arch-ring.  Then  in  Fig.  66  let  abc,  etc.,  be  these 
points  of  division  ;  connect  them  by  the  straight  lines  ab,  be,  cd, 
etc. ;  then  abcde  is  one  half  of  the  equilibrium  polygon  which 
follows  (nearly)  the  axis  of  the  arch.  We  have  now  to  deter- 
mine the  relative  and  actual  magnitudes  of  P,/5,/5,,  etc.,  so 
that  the  points  a,  b,  c,  etc.,  will  not  be  changed  in  position. 

The  load  at  the  crown  can  be  determined  at  once  from  the 
assumed  dimensions  and  weights.  Lay  off  one  half  of  this 
load  as  shown  in  Fig.  66,  and  draw  56  parallel  to  ge  until  it 
cuts  the  horizontal  at  P\  draw  StS3,  etc.,  parallel  to  ed,  dc,  etc., 
respectively :  then  the  distances  PtP4P3Pt,  etc.,  cut  off  on  the 
vertical  are  the  required  values  of  the  loads  at  e,  d,  c,  b,  etc. 
A  few  trials  will  place  the  material  above  the  ring  so  that  these 
values  will  very  nearly  obtain. 

We  have  now  all  of  the  general  dimensions  of  the  structure 
from  which  the  actual  loads  at  abc,  etc.,  can  be  computed. 


THE  MASONRY  ARCH. 


227 


Taking  abcde  as  the  axis  of  the  arch,  and  assuming  the  above 
loads  applied  at  the  points  abcde,  the  actual  values  of  //,,  Vlt 
and  Ml  can  be  found  by  means  of  the  formulas  already  demon- 
strated, and  the  true  equilibrium  polygon  drawn. 


FIG.  66. 

If  lead  joints  are  employed  at  the  skew-backs  and  the 
crown,  the  values  of  //,,  V^  etc.,  can  be  found  under  the 
assumption  that  the  arch  has  three  hinges,  trials  being  made 
under  the  assumption  that  the  hinges  lie  within  the  middle 
third  of  the  arch-ring. 

If  an  actual  hinge  is  placed  at  the  crown,  the  starting-point 
of  the  equilibrium  polygon  is  fixed. 

If  no  hinges  are  assumed,  then  the  starting-point  of  the 
polygon  must  be  determined  in  the  same  manner  as  for  the 
metal  arch. 

Extent  of  Loading  which  will  cause  the  Equilibrium  Polygon 
to  follow  the  Axis  of  the  Arch. — In  Fig.  66  let  mg'  represent 
the  load  at  g ;  then  at  the  joints  et  d,  c,  etc.,  lay  off  upwards  from 
the  lower  limit  of  the  arch-ring  the  loads  P^P^  etc.,  and  draw 
the  curve  jkm.  This  represents  very  nearly  the  upper  limit  of 
a  homogeneous  load  corresponding  to  the  polygon  abc,  etc. 
If  now  nop  is  drawn  parallel  to  jkm  at  a  distance  mp  below  this 


228  A    TREA  TISE   ON  ARCHES. 

curve,  the  shaded  portion  between  the  curve  nop  and  the  upper 
limit  of  the  arch-ring  represents  the  relative  amount  of  material 
to  be  placed  in  the  lateral  walls. 

If  the  live  load  over  all  is  included  with  the  dead  load,  the 
point  m  would  be  raised  an  amount  proportional  to  the  added 
live  load  measured  in  masonry  units. 

The  axis  of  the  arch  shown  in  Fig.  66  is  circular.  If  the 
angle  at  the  centre  had  been  larger  and  the  curve  jkm  continued, 
we  would  have  found  the  distance  between  it  and  the  arch-ring 
increasing  quite  rapidly  beyond  an  angle  of  45°  or  50°  from 
the  crown  and  becoming  infinite  for  the  semicircular  arch. 

For  this  reason  it  is  customary  to  consider  the  arch-ring  to 
act  as  an  arch  for  only  about  45°  or  50°  from  the  crown,  the 
masonry  in  the  abutments  or  piers  being  built  solid  in  horizontal 
courses  up  to  this  point. 

Moving  Load. — There  remains  now  to  be  determined  the 
effect  of  the  moving  load.  If  the  actual  maxima  stresses  are 
desired,  the  best  method  of  procedure  is  to  determine  the 
effect  of  each  load  or  concentration  independently  and  combine 
the  results.  In  most  cases,  however,  the  effect  of  the  moving 
load  is  small,  and  it  is  necessary  to  consider  but  two  cases, 
namely,  moving  load  over  all  and  moving  load  extending  from 
one  support  up  to  the  crown. 

Change  in  Dimensions. — If  after  trial  it  is  found  that  some 
equilibrium  polygon  for  dead  and  live  load  combined  departs 
from  the  middle  third  of  the  ring,  the  depth  of  the  ring  may 
be  changed ;  this  need  not  necessitate  a  new  calculation  unless 
a  great  change  is  made,  for  the  effect  of  the  added  material  is 
likely  to  be  very  small,  especially  if  the  equilibrium  polygon 
for  the  dead  load  follows  the  axis  of  the  arch-ring.  In  case 
the  equilibrium  polygon  lies  outside  of  the  middle  third  at 
any  section,  it  does  not  necessarily  make  the  structure  unsafe 
unless  the  intensity  of  the  pressure  is  sufficient  to  crush  the 
material.  The  joints  may  open  a  little  on  the  side  farthest 
away  from  the  polygon,  so  that  it  is  not  good  policy  to  so 
design  the  ring  that  there  is  any  such  tendency. 

Concrete  and  Brick  Arches. —  Evidently  concrete  and  brick 


THE   MASONRY  ARCH.  229 

arches  can  be  designed  in  the  manner  outlined  for  the  stone 
arch,  using  proper  judgment  as  to  the  strengths  of  the  materials. 
The  concrete  arch  may  even  be  made  lighter,  since  it  has 
considerable  strength  in  tension. 


ARCHES  WITH  LEAD  IN  THE  JOINTS  AT  THE  SPRINGING  AND 
THE   CROWN.* 

In  order  to  reduce  as  much  as  possible  the  uncertainty  of 
the  location  of  the  equilibrium  polygon  at  the  crown  and  the 
springing,  and  also  to  reduce  to  a  certainty  its  location  within 
limits,  German  engineers  have  placed  lead  in  the  middle 
thirds  of  the  joints  specified.  Evidently  the  equilibrium 
polygon  cannot  lie  far  outside  of  the  middle  third  at  these 
joints,  as  the  lead  acts  similarly  to  a  hinge.  After  the  false- 
works have  been  removed  the  masonry  adjusts  itself  until 
every  joint  is  in  equilibrium.  Nearly  all,  if  not  all,  this 
adjustment  takes  place  at  the  lead  joints,  which  are  com- 
pressed in  thickness  and  expanded  around  the  edges  until  the 
pressure  per  square  inch  does  not  exceed  about  3500  pounds. 

German  engineers  design  these  lead  joints  so  that  the 
maximum  intensity  of  the  pressure  does  not  exceed  about 
1600  pounds  per  square  inch,  and  have  been  very  successful 
in  their  application  of  the  method.  After  the  structure  is 
about  completed  and  the  entire  dead  weight  is  in  place  the 
joints  at  the  springing  are  filled  with  cement  and  the  arch 
becomes  fixed  at  the  ends  for  any  additional  loads. 


*"  Fonts  en  Magonnerie  avec  Articulations  a  la  clef  et  au  joint  de  Rup- 
ture." Par  M.  G.  La  Riviere.  Annales  da  Fonts  et  Chaiisse'es,  juin,  1891. 
Abstract,  Engineering  News,  Oct.  24,  1891. 


230 


A    TREATISE   ON  ARCHES. 


ARCHES  WITH  STEEL  OR  IRON  PINS  AT  THE  CROWN  AND  THE 
SKEW-BACKS. 

Recently  there  has  been  constructed  in  Switzerland  a 
concrete-arch  bridge  which  has  articulations  at  the  springing- 
joints  and  at  the  crown  composed  of  convex  steel  bearings 
resting  in  concave  steel  sockets  or  grooves.  The  entire  depth 
of  the  arch-ring  is  reinforced  with  metal  and  the  steel  bearings 
placed  at  the  centres  of  the  joints. 

This  mode  of  construction  definitely  fixes  the  equilibrium 
polygon  at  the  springing-joints  and  the  crown. 


EARTH-FILLED    SPANDRELS 


In  small  arches  the  spandrels  are  often  filled  with  earth 
from  the  arch-ring  up  to  the  roadway. 


FIG.  67. 

Assuming  the  earth  to  produce  only  the  vertical  pressures 
upon  the  arch-ring  due  to  its  weight,  the  determination  of  the 
equilibrium  polygon  offers  no  especial  difficulties.  But  prob- 
ably the  earth  causes  other  than  vertical  forces,  and  these  are 
more  or  less  indeterminate. 

If  the    earth  is  assumed  to  be  a  homogeneous  granular 


*  The   Coulouvreniere   Concrete-arch  Bridge,  Geneva,   Switzerland. 
gingering  News,  Aug.  6,  1896. 


En- 


THE  MASONRY  ARCH.  2$l 

mass,  then  the  pressure  upon  the  arch-ring  at  any  point  can 
be  fairly  well  determined  from  the  Theory  of  Earth-pressure.* 

If  the  arch  is  designed  for  this  earth-pressure,  it  will  sup- 
port a  very  considerably  increased  load  at  the  crown,  owing  to 
the  resistance  of  the  earth  over  the  haunches  against  heaving. 

Another  feature  which  places  the  method  of  considering 
the  earth-pressure  acting  against  the  ring  as  against  a  retain 
ing-wall  upon  the  safe  side  is  that  longitudinal  side  walls 
must  be  used  to  retain  the  earth  in  the  spandrels.  These 
walls  undoubtedly  relieve  the  arch-ring  from  the  direct  thrust 
of  the  earth. 

If  a  retaining-wall  is  placed  over  the  abutments,  then  the 
earth-filling  may  as  well  be  treated  as  a  vertical  weight  upon 
the  arch-ring. 

PART    EARTH    AND    PART    MASONRY    SPANDREL-FILLING. 

Under  the  assumption  of  vertical  loading,  it  is  found  often 
that  spandrels  filled  with  earth  alone  are  too  light  to  cause  the 
equilibrium  polygon  to  follow  the  axis  of  the  arch;  then  the 
spandrels  are  partially  filled  with  masonry,  as  shown  in  Fig. 
68. 


EARTH 


FIG.  68. 


This  masonry  is  usually  concrete  or  rubble  masonry.      It 
is  seldom  of  the  same  class  as  the  arch-ring  masonry. 

As  constructed,  the   upper  limit   of  this   masonry  filling 

*  Retaining  Walls  for  Earth,  by  M.  A.  Howe  ;  John  Wiley  &  Sons,  N.  Y. 


232  A    TREATISE   ON  AXCHES. 

slopes  very  gradually  from  the  crown  towards  the  skew-backs; 
hence  the  horizontal  thrust  of  the  earth  above  is  practically 
eliminated. 

The  exact  action  of  this  spandrel-filling  upon  the  arch-ring 
is  indeterminate.  The  assumption  that  it  acts  as  vertical 
forces  is  on  the  safe  side. 

MASONRY    SPANDRELS   WITH    LONGITUDINAL   VOIDS. 

Here  the  haunches  are  lightened  by  running  longitudinal 
walls  above  the  arch-rib  and  connecting  them  by  arches  or 
slabs  immediately  below  the  roadway,  as  shown  in  Fig.  69. 


FIG.  69. 

The  amount  of  space  to  be  left  void  can  be  found  by  the 
method  outlined  for  lateral  voids,  but  the  masonry  un- 
doubtedly exerts  a  much  less  pressure  upon  the  arch-ring 
than  under  the  assumption  of  vertical  loads.  Just  what  the 
pressure  is  cannot  be  determined.  Such  arches  seldom  if 
ever  fail  at  the  haunches  owing  to  the  resistance  offered  by 
the  solid  longitudinal  spandrel-walls. 

If  the  arch-ring  is  designed  for  vertical  loads  the  crown 
will  not  rise,  as  these  walls  cannot  possibly  exert  a  pressure 
equivalent  to  their  weight.  In  fact  good  masonry  can  be 
stepped  at  an  angle  of  at  least  50°  from  the  horizontal  and  be 
perfectly  stable,  provided  the  weight  is  balanced  over  the  pier 
or  abutment. 


THE  MASONRY  ARCH.  233 


ARCHES    HAVING    SPANS    LESS    THAN    TWENTY-FIVE    FEET. 

These  can  be  proportioned  in  the  manner  outlined  for 
larger  arches,  but  usually  the  ring  is  made  much  deeper  than 
necessary  owing  to  the  economy  in  using  material  of  certain 
dimensions.  Stone  arches  seldom  have  ring-stones  less  than 
one  foot  deep. 

We  have  pointed  out  some  of  the  difficulties  which  arise 
in  the  consistent  designing  of  masonry  arches  of  the  usual 
type.  The  principal  difficulty  appears  to  be  the  determina- 
tion of  the  magnitudes  and  directions  of  the  forces  due  to  the 
dead  load.  If  these  forces  are  assumed  as  acting  vertically 
and  in  magnitude  the  weight  of  the  material  included  between 
vertical  planes  then  the  arch  can  be  designed  by  the  formulas 
already  deduced  for  elastic  arches,  or  by  the  direct  and  very 
consistent  method  proposed  by  Alexander  and  Thomson, 
which  we  will  explain  in  the  following  pages.  For  the 
assumptions  made,  this  method  is  the  most  general  and  con- 
sistent which  has  been  advanced  up  to  the  present  time. 


CHAPTER  XIIL 

ALEXANDER  AND  THOMSON'S   METHOD   FOR  DESIGNING 
SEGMENTAL  MASONRY  ARCHES.* 

"  THE  Transformed  Catenary  is  shown  by  Rankine  (Civil 
Engineering,  Art.  131)  to  be  the  form  of  equilibrium  for  an 
ideal  linear  rib  or  chain  under  the  uniform-vertical-load  area 
between  itself  and  a  horizontal  straight  line.  This  curve  has 
received  considerable  attention  from  early  times  because  of 
its  importance  in  designing  arches,  and  is  known  best,  per- 
haps, by  engineers  as  the  equilibrium  curve. 

"  It  seems  to  have  been  assumed  that  the  transformed 
catenary,  like  the  common  catenary  and  the  parabola,  had  its 
curvature  continuously  diminishing  from  the  vertex  out- 
wards. 

"  In  the  following  investigation  it  is  shown  that  a  very 
close  resemblance  exists  between  certain  of  these  equilibrium 
curves  and  the  circle  —  a  fact  important  to  engineers." 

EQUATION   OF  THE   COMMON   CATENARY. 
From  Rankine's  Civil  Engineering,  Art.  128, 


(I) 


*  Transactions  of  the  Royal  Irish  Academy,  vol.  xxix,  part  ill,  1888.  On 
Two-nosed  Catenaries  and  their  Application  to  the  Design  of  Segmental  Arches. 
By  T.  Alexander,  C.E.,  Professor  of  Engineering,  Trinity  College,  Dublin; 
and  A.  W.  Thomson,  B.Sc.,  Assoc.  Mem.  Inst.  C.E.,  Lecturer  in  the  Glasgow 
and  West  of  Scotland  Technical  College.  This  is  an  elaborate  paper,  contain- 
ing many  interesting  things  which  are  omitted  here  as  not  being  essential  for 
the  mechanical  method  of  designing  arches. 

234 


SEGMENTAL   MASONRY  ARCHES 


235 


where  y  =  the  ordinate  of  any  point ; 

x  •=•  the  abscissa  of  any  point  having  the  ordinate^ ; 

m  —  the  parameter  ; 
and       e  =  the  base  of  the  Naperian  system  of  logarithms. 


w  J          Directrix        N 


0      K 


FIG.  70. 

THE  TRANSFORMED   CATENARY. 

The  locus  of  a  transformed  catenary  is  obtained  by  in- 
creasing or  decreasing  all  the  ordinates  of  a  common  catenary 
by  a  given  ratio  r. 

Then  for  the  transformed  catenary 


dy 

tan  0  =  -r-  =  -\em  — 

where  y^  is  the  value  of  y  when  x  =  o 

P  = 

<*-y 

dx* 


m 
=  rm. 


my 

py_ 

m? 


.      (II) 
.  (Ill) 

(IV) 


A    TREATISE   ON  AKCHES. 


where    0   is   the   slope   at   any   point   and   p  the   radius    of 
curvature. 

For  the  crown  (iv)  becomes 


and  hence  (v)  becomes 


sec8  0  =  -«L 

PoJo 


(VI) 


(VII) 


THE  TWO-NOSED   CATENARY. 

An  investigation  of  (iv)  for  maxima  and  minima  shows 
that  for  values  of  r  less  —-=.  there  is  a  maximum  radius  of 
curvature  pt  at  the  crown  and  a  minimum  radius  of  curvature 

X  Q    a;, KO  Directrix 


FIG.  71. 

p,  at  a  pair  of  points  (-£?,/?/)  symmetrical  about  the  crown, 
where 


(VIII) 


Such  catenaries  are  called  two-nosed. 


SEGMENTAL   MASONRY  ARCHES.  2tf 

If  m  be  assumed  as  unity  and  r  be  given  values  less  than 
V^,  the  values  of  plt  y^  xlt  ya,  and  0,  can  be  readily  com- 
puted. A  large  number  of  these  values  are  given  in  Table  A. 

As  an  aid  in  computing  the  ordinates,  etc.,  of  the  two- 
nosed  catenary,  the  general  formulas  may  be  put  in  the 
following  forms  : 

Let 

,=•£=,.-*.£  =  *      .....       (ix) 
f»  ./a 

Then,  from  (vili), 


From  (III), 

1  ^; (xi) 


2    ,   -*) (XII> 


From  Rankine's  Civil  Engineering,  Art.  131, 


(XIII) 

It  may  be  noted  here  that,  given  a  certain  value  of  s,  all 
quantities  are  directly  proportional  to  m  excepting  0, ,  which 
is  constant  for  any  given  value  of  s,  regardless  of  any  change 
in  m  or  />,. 


238  A    TREATISE    ON  ARCHES. 

THE  DESCRIBED  CIRCLE. 

In  Fig.  71  if  B£i  be  prolonged,  it  will  cut  AQ,  in  Qt.  If 
Q1  be  taken  as  a  centre  and  BtQ,  as  a  radius,  and  a  circle 
described,  it  will  evidently  lie  wholly  above  the  two-nosed 
catenary  between  the  points  B,  and  Z?/.  This  circle  will  also 
lie  beyond  the  catenary  curve  for  some  distance  beyond  B^ 
and  B{,  cutting  it  finally  in  B4  and  B{. 

Let  RI  be  the  radius  of  the  described  circle.  Then,  from 
Fig.  71, 

R^  =  x^  cosec  0, , (XIV) 

OQl=  b  =y1  +^j  cos0,,      ....       (XV) 
and 


(xvi) 


The  values  of  j?,  ,  b,  and  F0  are  given  in  Table  A  for  the 
values  of  r  which  were  used  in  computing  the  ordinates,  etc., 
of  the  two-nosed  catenary. 

An  examination  of  this  table  shows  for  r  =  V%,  or  s  =  •£, 
that  j0  =  F0,  or  the  described  circle  touches  the  two-nosed 
catenary  at  the  crown.  That  is,  .#,/?,'  and  A  and  K  coincide. 
Also,  that  between  the  values  of  s  —  0.027  and  s  =  0.0204,  F0 
changes  sign,  indicating  that  the  described  circle  cuts  the 
directrix. 

The  distance  apart  of  the  described  circle  and  the  two- 
nosed  catenary  at  the  crown  is 

XA=*.=s.-Yt  ......    (xvii) 

The  values  of  60  are  given  in  Table  A. 

THE   THREE-POINT   CIRCLE. 

Evidently  for  the  two-nosed  catenary  there  must  be  a 
point  beyond  Bl  which  has  the  same  radius  of  curvature  as  at 


SEGMENTAL   MASONRY  ARCHES. 


239 


the  crown.     There  will  be  a  similar  point  on  the  opposite  side 
of  the  crown.     A  circle  passed  through  these  three  points  will 


Directrix 


evidently  lie  below  the  catenary  between  /?„  A,  and  /?/.     This 
circle  is  called  the  three-point  circle. 

Let  R^  be  the  radius  of  the  three-point  circle  which  passes 
through  the  three  points  of  equal  curvature  of  the  two-nosed 
catenary. 


,      (XVIII) 

y.        Vs 

y^  =  y9  sec8  <pt—mVs  sec3  0,. 

.     .         (XIX) 
(XX) 

sec'  0  -     /  !       3       X 

V  J      4       2 
tan*0.  —  A  I         ^  —  ^  . 

(xxi) 

V  j      4      2 
f      »      j^  ^an  ^A 

.    .     (xxii) 

/y  YTTT\ 

*  "~       &€\                f7  '   ' 

*(y.-y.)  ' 

240 


A    TREATISE   OX  ARCHES. 


The  values  of  ;ra,  ja,  02,  /?,  pa,  and  j?3  are  tabulated  in 
Table  A,  from  which  we  see  that  02  and  ft  differ  but  little  until 
s  =  0.027,  and  then  the  difference  is  but  2°  28',  so  that  in 
many  calculations  one  angle  can  be  used  for  the  other. 


RELATIVE  POSITIONS    OF  THE  DESCRIBED  AND  THREE-POINT 
CIRCLES 


For  Values  of  s 
from  -3  to  -02  tlie 
Circular  Arcs  Converge 


From  Table  A  the  distance  between  the  centres  of  the  two 
circles  is  found  to  be  very  small  when  compared  with  the 
length  of  the  radii,  so  that  the  angular  distance  of  the  nose  B9 
from  the  crown  is  sensibly  the  same,  whether  measured  on  the 
described  or  three-point  circle.  Then 


(xxv) 


and  approximately,  for  values  of  s  from  0.333  to  0.01, 

tf,  =  .ff,  -  X,  +  A  cos  0,      .     .    .       (xxvi) 


and 


where 


f,  =  KI  —  R*  +  A  cos  /?,...     (xxvii) 


(XXVIII) 


SEGMENTAL   MASONRY  ARCHES.  24! 

An  examination  of  the  values  of  Sg ,  3lt  and  dt,  given 
in  Table  A,  shows  that  for  values  of  s  from  0.333  to  0.027, 
tfa  >  6,  >  6y ,  or  the  two  circles  approach  each  other  as  they 
leave  the  crown. 

Between  s  =  0.027  and  s  =0.0204  (s  =  TV)»  ^0=^1=  ^,*  °r 
the  two  circles  are  concentric.  Beyond  these  values  of  s  the 
circles  diverge. 

Then  if  through  A  a  symmetrical  circular  arc  is  passed 
concentric  with  the  described  circle,  the  equilibrium  polygon 
or  two-nosed  catenary  will  lie  between  the  two  out  to  the 
points  of  rupture  B,  and  B£.  Then  much  more  will  it  lie  be- 
tween the  described  circle  and  one  of  a  less  radius  than  the 
concentric  circle. 

Table  B. — In  Table  A  are  given  the  various  co-ordinates 
of  points  on  the  described  circle  and  the  three-point  circle  for 
a  modulus  m  =  unity. 

Suppose  now  we  wish  to  base  all  of  these  quantities  upon 
the  radius  of  the  described  circle  and  take  its  value  as  unity, 
then  it  is  necessary  to  divide  each  linear  quantity  in  A  by  the 
corresponding  value  of  /?,. 

Table  B  is  the  result  of  such  an  operation.  The  object 
of  this  will  appear  from  the  following: 

Suppose  a  circle  of  radius  unity  be  drawn,  and  let  this 
circle  be  taken  as  a  described  circle ;  then  R^—\. 

Since  s,  0,,  and  0,  are  independent  of  Rlt  it  is  evident  that 
all  of  the  two-nosed  catenaries  in  Table  A  can  be  constructed 
within  this  circle  of  unit-radius  merely  by  changing  each 
ordinate  proportional  to  Rt.  We  have  then  to  the  scale  unity 
(Rl  =  unity)  an  exact  representation  of  the  relations  between 
the  several  curves  we  have  been  considering. 

If  Rt  has  any  other  value  than  unity,  we  have  only  to  mul- 
tiply these  quantities  by  the  new  value  of  the  radius. 

In  Fig.  73  let  XX  be  the  upper  limit  of  masonry  to  be 
supported  by  an  arch,  and  OA  the  depth  at  the  crown  when 
Rl  —  unity;  then,  from  Table  B,  OA  =  ya  can  have  values 
from  0.23  to  0.05,  R^  0S,  0,,  etc/,  corresponding  values,  and 


242  A    TREATISE   0A'   ARCHES. 

for  any  particular  case  the  equilibrium  polygon  wiii  never  be 
above  B^KB{  between  B^  and  j5/,  and  never  below  B^ABJ 
between  B^  and  B,  .  Then  if  the  portion  of  the  masonry 
between  B^KB^  and  B^AB^  were  cut  into  arch-stones  the 
structure  would  be  stable  under  the  assumption  that  B^AB^ 
does  not  sensibly  differ  from  BJB^AB^B^  which  is  the  case 
within  the  limits  of  our  values  of  s. 

If  this  arch  with  R,  =  unity  is  in  equilibrium,  then  any 
other  arch  of  the  same  proportions  would  be  in  equilibrium. 

But  although  we  have  equilibrium,  we  have  not  strength 
probably;  and  besides,  in  masonry  the  equilibrium  polygon 
should  not  depart  at  any  place  from  the  middle  third  of  the 
joints,  and  often  it  must  follow  more  closely  the  centre  line  to 
obtain  intensities  of  pressure  consistent  with  the  strength  of 
the  material  employed. 

The  arch-ring  specified  above  decreases  in  depth  as  it 
leaves  the  crown  while  the  pressures  upon  the  radial  joints 
increase,  indicating  that  the  lower  boundary  of  the  ring  should 
be  changed  so  that  the  depth  would  increase  as  the  stresses 
increase.  This  can  be  done  after  the  depth  at  the  crown  is 
known.  As  this  depends  upon  the  material,  we  must  deter- 
mine the  permissible  intensities  of  the  stresses,  etc. 

Horizontal  Thrust.  —  According  to  Rankine's  Civil  En- 
gineering, Art.  131, 

ffi  =  Hx  =  H  —  wm\    .     .     .     (xxix) 

where  w  is  the  weight  per  unit  volume  of  the  material  taken 
as  solid  from  the  directrix  down  to  the  two-nosed  catenary  or 
the  equilibrium  polygon,  or  approximately  the  three-point 
circle. 

If  d  is  the  depth  at  the  crown  from  the  directrix  to  the 
soffit, 


H  —  —u<  m*  =  wpji.  ....      (xxx) 


SEGMENTAL   MASONRY  ARCHES.  243 

If  T  is  the  thrust  at  any  point, 


(xxxi) 


Intensity  of  Pressure.  —  In  Chapter  I  it  was  shown  that  if 
the  resultant  pressure  on  any  rectangular  joint  was  applied  at 
the  third  point,  the  maximum  intensity  was  twice  the  average 
intensity  and  the  minimum  intensity  was  zero. 

Assuming,  then,  that  the  equilibrium  polygon  at  the 
crown  is  to  be  applied  at  the  lower  third  point  of  the  key- 
stone, if  t^  is  the  depth  of  the  key, 

/0  =  3<?o  .......     (XXXII) 

The  average  intensity  of  the  pressure  is 
H 


=77  =  —' 

and  the  maximum  intensity 


Let  the  safe  strength  of  sandstone  be  576008  pounds   per 
square  foot  and  w  =  140  pounds  per  cubic  foot;  then 

wpad  ,  5  76008 

2  —  —  must  not  exceed  —  -  — 


* 
—  —  must  not  exceed  205  ; 

hence  the  maximum  multiplier  which  can  be  used  in  Table  B, 

pnd 

for  sandstone  is  205  -.  --  —  . 


244  A    TREA  TISE   ON  ARCHES. 

Table  Bl  derived  from  Table  B.  —  The  values  of  s  remain 
the  same  as  in  Table  B. 


-£—  is  derived  from  taking  p0  from  Table  B  and  multiply- 

d 
ing  it  by  —  . 

The  maximum  multipliers  are  found  as  explained  above, 
upon  the  supposition  that  /„  is  never  less  than  one  foot  for 
stone  masonry  or  one  brick  for  brick  masonry,  nor  greater 
than  reasonable  dimensions. 

The  remaining  factors  depend  upon  R,  which  is  found  as 
follows: 

In  order  that  equilibrium  may  still  exist  under  the  addi- 
tional masonry  due  to  increasing  the  depth  at  the  crown  by 
£„,  and  that  the  arch-ring  may  increase  in  depth  as  it  departs 
from  the  crown,  the  limit  of  the  soffit  is  made  a  three-point 
circle  for  a  two-nosed  catenary  which  lies  the  distance  d  below 
the  directrix  at  the  crown.  Then  entering  Table  A  with 
y^  =  d,  the  corresponding  value  of  Ry  is  found  for  m  =  unity. 
If  this  value  of  R^  be  divided  by  the  corresponding  value  of  R,, 
the  result  will  be  the  value  of  R  given  in  Table  Bt  on  the  line 
containing  the  assumed  value  of  d. 

In  a  manner  quite  similar  other  supplementary  tables  are 
formed. 

ADVANTAGES   OF   THE   METHOD. 

One  of  the  principal  advantages  of  the  above  method  is 
that  any  uniform  load  over  the  entire  span  can  be  added 
without  changing  the  described  circle  which  is  the  boundary 
of.  the  kernel  of  the  arch-ring,  provided  the  added  load  is 
not  sufficient  in  its  equivalent  of  masonry  to  make  'yt  -t-  p0 
greater  than  one  third.  The  effect  of  the  added  load  is 


SEGMENTAL   MASONRY  ARCHES.  245 

merely  a  change  in  the  angle  0,,  which  grows  smaller,  and  a 
change  in  tf0,  which  also  grows  smaller,  thus  leaving  the 
equilibrium  polygon  within  the  kernel  designed  for  the  original 
mass  of  masonry. 

This  is  particularly  advantageous  in  the  case  of  moving 
loads. 

It  is  to  be  noted  that  the  depth  of  the  key  depends  not 
only  upon  equilibrium,  but  upon  the  strength  of  the  material, 
and  that  this  depth  is  a  function  of  j0  and  <?0,  thereby  securing 
a  keystone  which  varies  consistently  with  different  conditions. 

Again,  for  given  conditions  there  is  a  perfectly  definite 
form  and  size  of  arch,  which  can  be  obtained  without  any  of 
the  usual  cut-and-try  methods. 

Since  for  a  solid  load  having  the  horizontal  directrix  for 
the  upper  limit  the  lower  limit  is  a  two-nosed  catenary  when 
equilibrium  exists,  and  since  within  the  limits  of  our  tables  an 
indefinite  number  of  two-nosed  catenaries  can  be  constructed 
having  different  values  of  y^  it  must  follow  that  the  homo- 
geneous material  between  any  two  two-nosed  catenaries  of  the 
same  family  (that  is,  transformed  from  the  same  common 
catenary)  must  be  in  equilibrium,  or  any  combination  of  such 
areas.  For  such  conditions  the  weight  of  the  material  may 
be  taken  as  the  average,  and  then  reduced  by  the  ratio  of  the 
loaded  area  to  the  total  area  between  the  two-nosed  catenary 
forming  the  soffit  and  the  directrix. 

The  three-point  circle  can  always  be  used  for  the  limits  of 
the  loading  since  it  differs  so  little  from  the  two-nosed 
catenary  within  the  limits  of  our  tables. 

Thus  almost  any  kind  of  spandrel-filling  can  be  used  pro- 
vided its  upper  limits  are  always  three-point  circles,  members 
of  the  same  family  as  the  line  of  stress. 


UNSYMMETRICAL   MOVING   LOAD. 

Since  the  moving  load  is  usually  small  in  comparison  with 
the  dead  load,  unsymmetrical  loading  need  not  be  considered. 


246  A    TREA  TISE   ON  AKCHES. 

For  a  moving  load  covering  but  one  half  of  the  span  the 
equilibrium  curve  at  the  crown  is  raised  a  little,  thus  leaving  a 
small  distance  between  it  and  the  dead-load  curve  from  the 
unloaded  portion.  The  effect  of  this  is  a  couple  tending  to 
turn  the  key.  This  turning  can  be  prevented  by  the  masonry 
filling,  as  illustrated  in  Example  6. 


CHAPTER   XIV. 

EXAMPLES  ILLUSTRATING  ALEXANDER  AND  THOMSON'S 

METHOD   FOR   DESIGNING   SEGMENTAL    MASONRY 

ARCHES.* 

Ex.  i°.  Design  of  a  sandstone  segmental  arch  with  ver- 
tical load:  span  75  feet  and  depth  of  surcharge  at  crown 
about  i  foot  4  inches.  The  springing  to  be  the  joint  of 
rupture. 

Here  2c  =  75  and  d  —  t0  =  i£;  their  ratio  is  56.25.  We 
find  by  trials  on  Table  B,  that  2c  -f-  (d  —  t^)=  53  occurs  on 
the  line  where  s  =  0.05,  and  the  multiplier  required  on  that 
line  to  make  2c  into  75  is  50.07,  about  half  the  maximum 
multiplier  given  under  sandstone  in  the  table;  so  we  shall 
have  a  factor  of  safety  of  about  twice  ten. 

From  Table  Bt  we  obtain  the  relative  values  given  below, 
and  multiplying  them  by  50.07  we  obtain  the  absolute  values. 

s         Mult.  d  t0  t-t  R  k  2c 

0.089      0.06 1       O.I23      0.869      0-427      1.498 


}'°7     ^4.46       3.05       6.15    43.5       21.4          75  feet. 

The  radius  and  rise  of  soffit  are  43.5  and  21.4  feet;  the 
thickness  of  arch-ring  at  the  crown  and  springing,  3  feet  o 
inches  and  6  feet  2  inches;  the  surcharge  being  1.41  feet  or 
nearly  I  foot  4  inches,  as  required. 

From  Table  B,  for  s  =  0.05  we  have 

s  Mult.  y?,  PO  Yo  <5,  <5a 

(        I  1.3634      0.0478      0.0478      0.0167 

0.05     50.07     j  50  07  68  265       2.393        1.02         0.836  feet. 

*  Examples  i  to  7  inclusive  are  from  Alexander  and  Thomson's  paper. 

247 


248  A    TREATISE   ON  ARCHES. 

At  the  crown  the  thrust  on  the  arch-ring  per  foot  of 
breadth  is  H  =  wp^d  —  140  X  68.265  X  4.46  =  42,600 
pounds;  the  average  intensity  of  the  stress  is  42,600  -=-  3.05 
=  14,000  pounds;  and  hence  the  maximum  intensity  is  2  X 
14,000  =  28,000  pounds  per  square  foot,  giving  a  factor  of 
safety  of  576,000  -f-  28,000  =  20. 

At  the  springing  T  —  H  sec  0,  =  84,000;  the  average 
stress  is  84,000  -4-6.15  =  13,700;  and  since  the  deviation  of 
the  stress  is  \t^  —  tfa  =  x0  =  1.025  —  0.836  =  0.189  above 
the  centre  of  the  joint,  then  from  Chapter  I,  page  9, 

f     ,   &r0\  (     ,   6(o.i89)\ 

max.  intensity  =  /.  ^i  +  -£)  =  13700(1  +    \l^i) 

=  16,200  pounds  per  square  foot, 

which  gives  a  factor  of  safety  of  576,000  -=-  16,200  =35. 

2°.  If  a  live  load  of  220  pounds  per  square  foot  of  road- 
way be  placed  upon  the  structure  (Ex.  i°),  find  the  new  line 
of  stress  in  the  arch-ring  and  the  intensities  of  stress  at  the 
crown  and  springing. 

The  height  of  superstructure  equivalent  to  this  live  load 
is  h  =  220  -T-  140  =  1.571  feet  of  sandstone.  Here  we  have 
to  find  a  new  two-nosed  catenary  still  inscribed  in  the  same 
circle,  Rl  =  50.07,  forming  the  upper  boundary  of  the  middle 
third  of  the  arch-ring,  as  already  designed  (Ex.  i°),  but  to  a 
directrix,  h,  higher  than  before.  Adding  //  to  the  old  value 
of  F0  we  get  2.393+  1.571  or  3.964,  which,  divided  by 
Rl  =  50.07,  gives  us  0.0793  as  a  new  relative  value  of  Y,  , 
which  is  found  in  Table  B  at  the  line 

*  <fra  mult.  /?,  Po  F0  «0  8, 

I  1.2384        0.0793         O.OI34 


0.075        55    3        50.07 

This  is  a  new  two-nosed  catenary,  of  a  different  modulus 
and  of  a  different  family,  so  that  the  soffit  already  designed 
will  not  be  mathematically  the  three-point  circle  of  another 
member  of  the  family  of  this  line  of  stress,  but  it  will  sensibly 
be  so.  The  joints  of  rupture  have  gone  up  to  5  5°  3';  but  this 


SEGMENTAL   MASONRY  ARCHES.  249 

is  immaterial,  as  the  line  of  stress  is  now  closer  to  the  upper 
boundary  of  the  kernel,  and  will  therefore  be  wholly  in  the 
kernel  down  to  59°  31',  the  springing-joint. 

At  the  crown  now  we  have  the  thrust  H'  =  wp(d-{-  h)  = 
140  X  62.007(4.46  -j-  1.57)  =  5 2, 300  pounds;  average  inten- 
sity =  17,200  and  maximum  intensity  22,600  pounds,  being 
less  than  the  maximum  intensity  for  the  dead  load  alone, 
because  of  the  centre  of  stress  being  much  nearer  the  centre 
of  the  joint. 

At  the  springing  T'  =  H'  sec  59°  31'  =  103,200.  x  = 
0.539  feet  above  the  centre  of  the  joint,  and  the  maximum 
intensity  of  pressure  is  25,600  pounds  per  square  foot,  giving 
for  the  live  load  on  the  structure  a  factor  of  safety  of  about  22. 

3°.  Let  the  live  load  of  2°  cover  but  one  half  of  the  span; 
find  the  horizontal  thrust  to  be  balanced  by  the  backing  of 
the  voussoirs. 

For  dead  load  alone,  H  =  42,600  pounds; 

For  live  and  dead  loads,     H  =  52,300      " 
Hence  the  required  thrust  is  =    9,700      " 

4°.  Fig.  74.  Suppose  the  arch-ring  spandrels,  etc.,  of  1° 
have  by  means  of  voids  in  the  superstructure  an  average 
weight  of  100  pounds  per  cubic  foot.  Find  results  corre- 
sponding to  those  of  i°. 

For  stability,  and  to  give  the  required  value  of  d  —  t0 ,  the 
dimensions  in  i°  are  required,  just  as  before,  but  the  stresses 
will  be  altered  in  the  ratio  140  :  100.  H  now  becomes  about 
30,500  pounds  and  T 60,000  pounds,  giving  factors  of  safety 
of  28  at  the  crown  and  49  at  the  springing. 

The  voids  should  be  so  arranged  that  their  boundary  may 
be  roughly  a  member  of  the  same  family  as  the  line  of  stress, 
by  making  the  ordinates  of  the  boundary  a  constant  fraction 
of  those  of  the  soffit. 

This  should  be  done  when  the  spandrels  are  partially  filled 
with  masonry  and  then  the  remainder  with  earth. 

5°.  Fig.  74.  Let  a  live  load  of  157  pounds  per  square 
foot  of  roadway  be  over  the  whole  span  of  the  bridge 


250 


A    TREATISE   ON  ARCHED 


(Ex.  4°).     Find  the  line  of  stress  and  the  intensity  of  the 
stress  at  crown  and  springing. 


tve! 


oc     a 
<      5 


I  S 

5  § 

Q  o 

I  A 

5  * 

.  ..i  - 


Hs 

SE 


Z  « 

o  y 

OT  5 

S  H 


The  equivalent  height  of  structure  is  1.57  feet  taking  the 


SEGMENTAL   MASONRY  ARCHES.  2$I 

new  density  into  account,  so  that  the  solution  is  the  same  as 
2°,  only  we  must  alter  the  quantities  in  the  ratio  140  :  100. 

H '=  37,500  nearly,  and  T=  73,700  nearly. 

6°.  Fig.  74.  Let  the  live  load  in  5°  be  over  only  one 
half  the  span.  Find  the  amount  of  horizontal  thrust  to  be 
balanced  by  the  frictional  stability  of  the  vault-covers  butting 
against  the  higher  voussoirs.  Find  also  the  distance  back  to 
which  the  vault-covers  must  extend  to  balance  it. 

The  thrust  P  —  37,500  —  30,500  =  7000  pounds  per  foot 
of  breadth.  If  the  under  side  of  the  vault-covers  come  up  to 
the  level  of  the  crown  of  the  soffit,  then  the  weight  per  foot 
of  breadth  of  bridge  on  the  spandrels  due  to  the  vault-covers, 
and  the  dead  load  over  them  alone,  is  wdL  =  140  X  4-46Z- 
=  624^.  Taking'  the  coefficient  of  friction  as  0.7,  then 
0.7  X  624^  =  7000,  or  L  =  about  16  feet.  The  voussoirs 
near  the  keystone  should  have  square-dressed  side-joints  until 
the  sum  of  their  vertical  projection  is  /„,  so  as  to  receive  the 
horizontal  thrust  of  the  vault-covers  truly;  the  spandrel  walls 
must  be  built  up  to  the  height  of  the  soffit  for  a  distance 
equivalent  to  16  feet  of  vault-covering,  when  they  may  be 
stepped  down. 

7°.  Design  of  a  semicircular  arch-ring  of  common  sand- 
stone, the  span  to  be  100  feet,  and  a  surcharge  of  at  least  i£ 
feet  being  required  for  the  formation  of  the  roadway,  laying 
of  gas-pipes,  etc.  The  data  are  R  =  50,  and  R  -j-  (d  —  /„) 
not  to  be  less  than  33.  On  Table  B,  the  lines  above  that 
with  s  =  .08  (in  order  to  make  R  into  50)  require  a  multiplier 
greater  than  the  maximum  given  for  sandstone;  these  lines 
are  therefore  excluded  on  the  question  of  strength,  while  the 
lines  below  that  with  s  =  .05  do  not  give  R  -~  (d —  /„)  so 
great  as  33,  and  are  excluded  by  requirements  of  the  road- 
way. Those  two  limiting  lines  give 
j  Mult.  R  d  t°  02  Factor  of  Safety. 

.08      53.6      50       5-9        2        54°  14'        575*6°  =  10.6 
.05      57-5      50       5.1       3-5       59°  3i'  ='8 


252  A    TREATISE   ON  ARCHES. 

The  upper  gives  greatest  economy  of  material  in  arch-ring, 
which  is  only  2  feet  at  crown,  but  less  economy  of  material  in 
superstructure,  as  d  is  larger,  and  also  less  economy  of  solid 
backing,  which  has  to  be  built  to  a  joint  5°  higher.  Hence 
the  line  midway  between  them  would  be  most  suitable  all 
round.  For  a  single  arch  a  line  a  little  nearer  the  upper  may 
be  adopted;  and  for  a  series  of  arches  a  line  nearer  the  lower, 
that  is,  in  favor  of  a  heavier  arch-ring  to  withstand  the  shocks 
transmitted  from  arch  to  arch.  The  best  lines  then  are 

s          Mult.         R         d        t0         /„  0.,  Factor  of  Safety. 

Single  arch 07      54-526      50      5.6     2.4      4.4      55°  53'       9  x  IO=  I2  ? 

Series  of  arches.   .06      55.804      50      5.3     2.9      5.4      57°  38'     — — -^=15.2 

55-8 

Compare  Rankine's  empirical  rule,  Civil  Engineering,  Art. 
290,  giving 

X  50 

2.92  respectively. 

The  solid  backing  must  be  brought  up  to  the  point  where 
the  joint  at  0,  meets  the  back  of  the  arch-ring,  and  below 
that  joint  the  arch-ring  may  be  of  the  uniform  thickness  /,. 
The  superstructure  may  readily  be  reduced  by  voids  and  the 
employment  of  material  of  less  density  than  sandstone,  till 
the  average  density  of  'the  whole  is  a  fifth  less  than  that  of 
sandstone,  which  would  raise  the  factors  of  safety  at  crown  to 
1 6  and  19.  The  factors  of  safety  at  joint  of  rupture  are  even 
greater  as  the  centre  of  stress  is  nearer  the  centre  of  the  joint, 
and  /,  -T-  /,  >  sec  #a.  By  means  of  the  values  obtained  for 
Po »  <^>  ^j »  the  thrust  at  crown  and  joint  of  rupture  and  the 
centre  of  stress  at  joint  of  rupture  are  calculated,  as  in  pre- 
ceding example.  A  tangent  from  this  last  point  enables  a 
suitable  abutment  to  be  designed. 


CHAPTER  XV. 
TESTS  OF  ARCHES. 

RECENTLY  the  Austrian  Society  of  Engineers  and  Archi- 
tects have  published  a  report  of  a  series  of  tests  made  upon 
full-size  arches.  The  publication  contains  131  folio  pages 
with  27  plates.*  The  experiments  are  minutely  described 
and  thoroughly  discussed,  and  a  comparison  made  between  the 
results  and  those  theoretically  obtained. 

The  tests  of  greatest  interest  were  those  made  upon  five 
arches  having  a  span  of  75. 4  feet,  a  clear  rise  of  15.1  feet,  and 
a  width  of  6.56  feet.     These  arches  were — 
i°.   Rough  quarry-stone; 
2°.   Brick; 
3°.   Concrete; 
4°.  Concrete  Monier  type; 
5°.   Steel. 

Rough  Quarry-stone  Arch. — This  arch  was  constructed  of 
rough  quarry-stone  laid  in  Portland  cement-mortar  composed 
of  i  part  cement  and  2.6  parts  sand,  the  test  being  made  51 
days  after  its  completion. 

The  thickness  at  the  crown  was  23.6  inches  and  at  the 
skew-backs  43.3  inches. 

The  loading  was  applied  vertically  at  five  points,  dividing 
the  half-span  into  five  equal  parts. 

The  ultimate  load  causing  rupture  was  660  pounds  per 
square  foot  over  one  half  of  the  span. 

*  Bericht  des  Gewolbe-Ausschusses  des  Oesterreichischen  Ingenieur-  und 
Architekten-Vereins.  Vienna,  1895.  See  also  Eng.  News,  Nov.  21,  1895, 
p.  351,  and  April  9,  1896. 

253 


254  A    TREATISE   ON  ARCHES. 

The  arch  failed  by  radial  cracks  appearing  on  the  extrados 
near  the  skew-backs  on  the  loaded  side  and  over  the  haunches 
on  the  unloaded  side. 

Brick  Arch. — This  arch  was  identical  in  dimensions  with 
the  stone  arch,  and  failed  in  a  similar  manner  under  a  load  of 
602  pounds  per  square  foot  over  one  half  of  the  span. 

Concrete  Arch. — The  thickness  at  the  crown  was  27.6 
inches,  and  at  the  skew-backs  27.6  inches. 

This  arch  was  made  up  of  segments  of  concrete  composed 
of  mixtures  of  different  proportions,  and  at  the  skew-backs  the 
joints  between  the  arch-ring  and  the  abutments  were  filled 
with  asphalt  about  \  inch  thick. 

The  arch  failed  under  a  load  of  742  pounds  per  square 
foot  over  one  half  of  the  span. 

Monier  Arch. — Here  the  general  dimensions  were  the 
same  as  before,  but  the  thickness  at  the  crown  was  only  13.8 
inches  and  at  the  skew-backs  23.6  inches. 

The  arch  failed  under  a  load  of  1300  pounds  per  square 
foot  over  one  half  of  the  span,  failing  by  cracking  as  follows: 

1st.   On  the  loaded  side  at  the  skew-back; 

2d.   On  the  unloaded  side  at  the  haunches;    and, 

3d.   On  the  loaded  side  at  the  haunches. 

Steel  Arch. — Failure  took  place  under  a  load  of  1564 
pounds  per  square  foot  over  one  half  of  the  span,  by  the 
buckling  of  the  unloaded  portion  near  the  haunches. 

Deformations. — Throughout  the  tests  careful  measure- 
ments were  made  of  all  deformations  caused  by  removing  the 
falseworks,  temperature  changes,  and  the  changes  in  loading. 

The  appearance  of  the  first  crack  was  noted,  with  the 
magnitude  of  the  load  causing  it. 

The  arches  were  finally  tested  to  destruction  and  the  load 
causing  failure  carefully  determined. 

From  these  records  a  comparison  was  made  with  theoreti- 
cal results. 

Comparison  with  Theory. — It  was  found  for  the  stone  and 
brick  arches  that  failure  occurred  in  the  joints,  the  mortar 


TESTS   OF  ARCHES.  2$$ 

separating  from  the  stone  or  brick.  The  adhesive  strength 
of  the  mortar  for  the  stone  arch  was  found  to  be  about  120 
pounds  per  square  inch  and  the  value  of  E  about  960,000. 

In  the  brick  arch  the  adhesive  strength  of  the  mortar  was 
about  70  pounds  per  square  inch,  and  the  values  of  E  varied 
from  340,000  to  470,000. 

From  the  results  of  the  tests  of  the  concrete  arch  the 
average  ultimate  strength  of  the  concrete  was  placed  at  290 
pounds  per  square  inch,  and  the  value  of  E  at  1,430,000. 

The  Monier  arch  cannot  be  discussed  theoretically,  owing 
to  the  use  of  metal  imbedded  in  the  concrete. 

The  value  of  E  as  determined  from  the  tests  of  the  steel 
arch  was  about  26,000,000,  which  is  a  little  smaller  than  the 
value  obtained  from  the  tests  of  small  specimens. 

Even  in  the  masonry  arches  the  deformations  were  pro- 
portional to  the  loads  up  to  a  certain  point,  showing  that  the 
material  behaved  the  same  in  the  arch  as  in  small  specimens 
for  testing. 

The  measuring  devices  placed  near  the  skew-backs  indi- 
cated that  on  the  loaded  side  the  arch  was  practically  fixed  at 
the  ends  and  on  the  unloaded  side  very  nearly  so.  Of  course 
the  concrete  arch  with  asphalt  plates  at  the  skew-backs  must 
be  excepted.  This  .  arch  behaved  neither  as  fixed  nor 
hinged,  and  the  theoretical  results  were  taken  as  the  mean 
of  those  obtained  by  considering  the  arch  as  fixed  and  then 
as  hinged. 

In  all  cases  the  arches  failed  at  points  which  theory  pre- 
dicted, 

CONCLUSIONS    DRAWN    FROM    THE     RESULTS     OF    THE     FIVE 
EXPERIMENTS. 

The  very  important  conclusion  drawn  from  these  experi- 
ments was  that  the  masonry  arches  behaved  very  nearly  as 
elastic  arches  fixed  at  the  ends,  and  hence  the  formulas  for 
elastic  arches  were  the  only  formulas  which  should  be  em- 
ployed in  designing  such  structures. 


256  A    TREATISE   ON  ARCHES. 

The  close  agreement  with  theory  under  the  method  of 
applying  the  loading  employed  in  these  experiments  is  a  very 
strong  argument  in  favor  of  the  type  of  spandrel  construction 
advocated  in  Chapter  XII.  A  few  very  old  bridges  and 
some  modern  bridges  have  been  constructed  after  this  form. 
The  only  argument  against  this  method  is  that  in  bridges  of 
long  span  the  effect  of  changes  in  temperature  sometimes 
cracks  the  masonry  above  the  small  arches;  but  this  can  be 
avoided  by  making  a  vertical  joint  near  the  skew-backs,  as 
was  done  in  the  Coulouvreniere  bridge. 


SPECIFICATIONS. 

The  following  specifications  are  advocated  in  Chapter  VII 
of  the  Austrian  report. 

All  Large  Arches  must  be  designed  according  to  the  Elastic 
Theory. — Two  cases  of  live  loading  may  be  considered:  (i°) 
load  covering  entire  span,  and  (2°)  load  covering  but  one  half 
of  the  span. 

For  railway  bridges  the  rails  should  be  at  least  3.28  feet 
above  the  crown  of  the  arch,  and  this  space  filled  with  some 
cushioning  material. 

Brick  and  stone  arches,  where  the  ratio  of  the  rise  to  the 
span  lies  between  one  half  and  one  fifth,  may  have  depths 
at  the  crown  as  specified  below. 

For  spans  of  30  metres,  thickness  of  crown  =  i.i  m. 
40        "  "  "  1.4  " 

«  it          55          <i  «  «  2.2    « 

So       "  "  "          2.7   " 

«          «      I00       «  «  «  ^  4  « 

"**>       "      120       "  "  "          4.1    " 

For  segmental  arches  the  thickness  at  the  skew-backs  may 
be  i£  the  thickness  at  the  crown,  and  for  semicircular  arches 
1.7  the  thickness  at  the  crown. 


TESTS   OF  ARCHES.  2  57 

The  width  of  the  bridge  at  the  crown  should  never  be  less 
than  the  following: 

For  spans  of  30  metres,  width  =  2.4  m. 

40        "  "           3.0  " 

"        65         »  "           4.5    « 

80        "  "           5.6   " 

«          «      IOo       "  "          7.0  " 

«          «      I20        <c  «          86  « 

If  the  width  at  the  crown  is  small,  the  width  at  the  skew- 
backs  should  be  one  twentieth  greater. 

In  all  cases  the  falseworks  should  be  as  rigid  as  possible, 
and  in  order  that  the  deformations  should  be  symmetrical  the 
arch  should  be  constructed  in  symmetrical  sections. 

TEST    OF    SMALL    ARCHES. 

In  connection  with  the  tests  mentioned  above  two  small 
arches  were  tested. 

M  outer  Arch. — This  arch  had  a  span  of  32.8  feet,  a  rise 
of  3.28  feet,  and  a  width  of  13.92  feet.  The  thickness  at  the 
crown  was  7.87  inches.  The  metal  gridiron  which  was  placed 
near  the  intrados  of  the  arch  was  made  of  pieces  0.39  inch 
and  0.27  inch  in  thickness,  the  former  running  longitudinally. 

The  spandrels  were  filled  even  with  the  crown  with  con- 
crete, which  carried  a  single  standard-gauge  railway  track. 

The  arch  was  first  tested  with  locomotives  covering  one 
half  of  the  span,  then  a  uniform  load  of  rails  was  placed  upon 
one  half  of  the  span. 

Cracks  appeared  near  the  springing  on  the  loaded  side 
under  a  load  of  920  pounds  per  square  foot.  The  arch  failed 
under  a  load  of  2010  pounds  per  square  foot  over  one  half  of 
the  span  by  the  yielding  of  the  abutments. 

Concrete  Arch. — A  concrete  arch  of  the  same  dimensions 
was  tested  six  months  after  being  built  and  no  signs  of  failure 
appeared  under  a  load  of  2110  pounds  per  square  foot  over 
one  half  of  the  span. 


2$8  A    TREA  TISE   ON  ARCHES. 


TESTS   OF   FLOOR   ARCHES. 

Austrian  Tests. — A  synopsis  of  these  tests  was  given  by 
Prof.  Merriman  in  Engineering  News,  April  9,  1896.  This 
synopsis  covers  the  ground  so  thoroughly  that  it  is  given 
below. 

Seventeen  arches,  having  spans  of  4.43  feet  and  8.86  feet, 
were  tested  to  destruction  by  a  uniform  load.  Of  these  four 
were  common  brick  arches,  five  were  of  special  forms  of  brick, 
three  were  concrete  arches,  three  were  Monier  arches,  one 
was  of  the  Melan  system,  and  two  of  corrugated  plates. 
Most  of  these  were  built  between  rolled  beams  in  the  manner 
usual  in  floor  construction,  these  beams  being  prevented  from 
spreading  by  plates  and  channels  at  the  ends,  and  also  by  a 
tie-rod  at  the  middle.  The  space  above  the  arch  and  between 
the  beams  was  levelled  up  with  earth,  upon  which  a  board 
floor  was  laid,  and  upon  this  pig  iron  was  piled.  The  tests 
were  made  four  months  after  the  arches  had  been  built.  All 
these  arches  were  designed  for  an  allowable  load  of  123 
pounds  per  square  foot  of  load,  besides  their  own  weight,and 
were  expected  to  rupture  with  about  eight  times  this  load, 
or,  say,  1000  pounds  per  square  foot. 

Seven  floor  arches,  with  spans  of  4.43  feet,  were  tested  in 
this  manner.  Under  a  load  of  1000  pounds  per  square  foot 
none  showed  cracks  or  signs  of  failure.  Under  1500  pounds 
per  square  foot  the  tests  of  two  arches  were  discontinued  on 
account  of  a  deformation  of  the  beams  and  their  connections, 
although  the  arches  themselves  were  intact.  On  the  other 
arches  the  load  was  increased  to  about  1650  pounds  per  square 
foot,  under  which  two  failed  and  three  remained  unbroken. 
In  each  case  the  deflection  of  the  crown  of  the  arch  was 
observed  for  different  loads:  under  1430  pounds  per  square 
foot,  for  example,  this  deflection  varied  between  0.39  and 
0.98  inch,  while  for  the  two  arches  that  failed  the  ultimate 
deflections  were  i.o  and  1.65  inch. 

The  conclusions  drawn  from  the  tests  of  these  small  floor 


TESTS   OF  ARCHES. 


259 


arches  are  as  follows:  (i)  That  common  brick  arches  4  43  feet 
in  span,  with  a  rise  of  0.44  foot,  and  laid  with  lime-mortar, 
show  such  slight  deformations  under  a  uniform  load  of  1430 
pounds  per  square  foot,  that  they  afford  ample  security  for 
all  common  buildings;  (2)  that  ring-courses  in  brick  arches  are 
preferable  to  longitudinal  courses;  (3)  that  beton  arches  3 
inches  thick,  made  of  I  part  of  Portland  cement  and  5  parts 
of  sand,  have  about  the  same  strength  as  brick  arches  6  inches 
thick;  (4)  that  flat  arches  give  a  much  higher  strength  than 
expected  (although  the  thrust  upon  the  floor-beams  is  of 
course  greater),  and  under  careful  construction  they  are  of 
ample  strength  for  all  architectural  purposes. 

A  second  series  of  tests  on  arches  of  8.86  feet  span  was 
conducted  in  a  similar  manner,  except  that  the  extra  uniform 
load  was  applied  only  over  one  half  the  span.  The  following 
table  gives  the  principal  data  regarding  these  arches,  as  also 
the  load  causing  rupture: 


Kind  of  Arch. 

Rise, 

in. 

Thickness, 
in. 

Dead  Load, 
Ibs. 

Applied  Load, 
Ibs.,  sq.  ft. 

j-28 

o  8 

CO- 

Brick    2                 .... 

a  q 

Corrugated  plate,  I.  .  .  . 
Corrugated  plate,  2.... 

9.8 
.       10.2 

2970 
2130 

974 

IIOO 

The  loads  in  all  cases  were  applied  gradually,  and  at  each 
increment  of  200  pounds  per  square  foot  the  vertical  and 
horizontal  displacement  of  the  crown  of  the  arch  was  meas- 
ured. The  concrete  arch  fulfilled  all  expectations,  and  its 
deformation  was  less  than  one  half  of  that  for  the  brick  arch. 
Of  the  two  Monier  arches,  the  first  was  built  between  rolled 
beams,  while  the  second  had  solid  concrete  abutments,  the 
effect  of  which  was  to  greatly  increase  its  strength.  The  first 
brick  arch  was  of  common  brick,  and  the  second  of  a  patent 
brick  of  much  less  thickness;  the  test  thus  shows  that  brick 


200  A    TREA  TISE   ON  ARCHES. 

less  in  thickness  than  the  common  kinds  should  not  be  em- 
ployed. The  first  corrugated-plate  arch  simply  butted  against 
the  floor-beams,  while  the  second  was  provided  at  the  ends 
with  angle-irons;  the  deflections  of  these  were  greater  than 
in  any  other  arch  except  the  brick  ones. 

A  third  series  of  tests  on  concrete  and  brick  arches  of  13.3 
feet  span  was  also -undertaken,  the  abutments  being  made  as 
nearly  immovable  as  possible.  A  brick  arch  of  13.9  inches 
rise  and  5.5  inches  thickness  was  loaded  over  half  the  span. 
Under  a  load  of  205  pounds  per  square  foot  the  vertical 
deflection  at  the  crown  was  0.29  inch,  and  the  horizontal 
movement  was  o.  1 1  inch.  When  the  load  reached  205 
pounds  per  square  foot,  a  small  crack  appeared  on  the  un- 
loaded extrados,  and  when  it  reached  275  pounds  per  square 
foot  rupture  occurred.  A  concrete  arch,  on  the  other  hand, 
cracked  at  410  pounds  per  square  foot  and  ruptured  at  663 
pounds  per  square  foot.  A  Monier  arch,  which  is  of  beton 
built  on  an  arched  network  of  heavy  wire  or  light  round  iron, 
cracked  at  512  and  ruptured  at  894  pounds  per  square  foot. 

The  Melan  system,  in  which  the  beton  or  concrete  is 
included,  between  arched  I  beams,  was  also  tested,  the  span 
being  4  metres,  or  13.1  feet,  the  rise  0.94  foot,  the  I  beams 
I  metre  apart,  and  the  thickness  3.15  inches.  This  arch 
was  loaded  on  one  side  up  to  1410  pounds  per  square  foot, 
when  the  test  was  discontinued  on  account  of  lack  of  pig  iron. 
Afterwards  an  area  of  I  metre  square  over  the  second  rib  was 
loaded  up  to  3360  pounds  per  square  foot,  when  failure 
occurred,  large  cracks  having  formed  at  3100  pounds  per 
square  foot.  This  test  shows,  of  course,  the  strength  of  the 
I  rib  rather  than  that  of  the  total  structure,  yet  there  can  be 
no  doubt  that  this  system  is  a  highly  efficient  one,  not  only 
for  floors,  but  for  small  bridges. 


APPENDICES. 


APPENDIX  A. 

INTEGRALS   EMPLOYED   IN   THE    DEDUCTION   OF  Ax  FOR 
PARABOLIC  ARCHES.     EQUATION  p(7g). 


Substituting  the  value  of  A$  as  given  in/(69), 
A<t>dy  =  yA^  +  -LF*  dy     2M,x  +  Vj?  -  H, 


Substituting  the  value  of  dy  —S-  -  dx  [from/(65)], 

P  •  •;    '01  i 

-L- 


-  6pb(X  -  «)].(£  -  x} 

263 


264  A    TREATISE   ON  ARCHES 

which  reduces  to 


From  (42), 

^,  =  F,  sin  0  +  7/,  cos  0. 
From/(59)> 


Then 


APPENDIX  A.  265 


n 

From./(6o),  m—~\  hence 


~    I    _V*  =  ^j    /    £ 

*/9          x  (  t/0 

From  (39)  and  (40), 


and 

V*  —  ^  ~  2P' /(40) 

Substituting  these  values  in  the  above  equation, 

-    f  ^dx="1^    /Tocos' 
EJo    F*  A  (Jo    L 

rr.  *  -,) 

+    /       //,  cos  <f)dx  —  2Q  cos1  <t>dx    > 

c/O       L  J) 


-  ^\_Pfa   cos'  ^  J  -  i  [^jf^08' 


i   ^i*  anc^  ^  =  —  pdz ;  hence  dy  —  —  zpdz,  and  we  have 


266  A    TREATISE   ON  ARCHES. 

Substituting  the  value  of  2, 

s'  0rfy  =  -  l-p  log 


f  I  2V 

In  practice  —  seldom  exceeds  -;  then  for  this  ratio  —  -^-  —  > 
=  —  ;  when  y  <  /,  -  -  <  —  .    Then  without  sensible  error 

we  may  take 


and 

/(74) 


/   cos2  <}>dx. 


or  *£  » 

As  before,  let  z  =  tan  0  = ;   then  cos*  0  = 


and  ^  =  —  pds,  and  we  have 

*  * 

/'cos'  0dfor  =  /    — T^dz  —  —  p\  tan"1  z  =  —  p\  <f>. 
I/O     I  — j—  2  \_ 

0  *. 

Therefore      ., 

sll0^=-/(0-00)=X</>o-0).      .     .    .    /(75) 

/'cos*  0^/7. 
. 

.        *  From  demoastration  by  Prof.  Weyrauch. 


APPENDIX  A.  267 


Therefore 


-    /   cos*  <£>dx.  - 
jT'cos'  $dx  =  [~=  -/ftan-1  ar  =  -/F0  =  —  /(0  —  «)• 

a  a  a 

Therefore 

jTcos'  0d^  =  X«  -  0)  .....    /(77) 

Substituting  /(74).  X75).  /(76).  and/(77)  in  /(73)».  reducing 
and  factoring,  we  obtain 


-  2P(y  -b}- 
Substituting  /(7i),/(72)»  and/(?8)  »n/(7o),  we  have 


268  A    TREATISE   ON  ARCHES. 


-  a)'  -f  3«5  -  15 


X79) 


APPENDIX   B. 

INTEGRALS    EMPLOYED   IN  THE  DEDUCTION  OF  Ay  FOR 
PARABOLIC  ARCHES.     EQUATION  ^(84). 


,,0   P»d 

t/O 

et°  f'dy  =  et 

t/O 
t/O 


-•«)] 

or 


269 


A    TREATISE   ON  ARCHES. 


e,  dx 


dx 

dy(  V*  sm  <t>+Hx  cos  0) 


=  —:  \  Vf  i    sin*  <f>dx  -j-  //^  I    cos1  <fidy. 

'          i/O  i/O 

From  (39)  and  (40), 

//,  =  //; -10 (39) 

and 

F,=  F.-ip; (40) 

hence 

/J^L      -  m  i        /*  •  •  /" 

"     *  tx  0  t/o 

»r»r«     .1    -r    r 

—  2   /*  /    sin  0d?.f     —  ^1  Q   I    cos'  0 

L     Ja  L      Ja 

/  sin"  0^  =    /  (i  —  cos4  (j>]dx  =  x  —  /(00  —  0); 

/*  ,-»* 

sin*  (fidx  =   /    (i  —  cos*  (f>}dx  =  x  —  a  —  t>(a  —  (b\ 
l'a 

r  py 

I      COS    0<//   =    ; -• 

Jo  P-\~  2J 


APPENDIX  B.  271 

Therefore 


-  ±P\x  -a-p(<*-  0)]  -  % 


Substituting.  /(8i),  p($2),   and  ^(83)   in  /(8o),  we  obtain 


APPENDIX  C. 
EFFECT  OF  THE  AXIAL  STRESS. 

To  illustrate  the  effect  of  the  axial  stress  we  will  solve 
several  examples  by  the  common  method  and  by  the  formulas 
which  take  into  account  the  axial  stress.  A  comparison  of 
the  results  thus  obtained  will  indicate  the  importance  of  this 
stress. 

In  the  following  examples  let  the  form  of  the  arch  be  para- 
bolic and  have  a  span  of  100.  Also,  let  the  (radius  of  gyra- 
tion)" =  4  =  m. 

ARCH  WITH  A  HINGE  AT  EACH  SUPPORT. 
(a)    Vertical  Loads. 

1°.  Assume  a  single  load  on  the  crown  of  the  arch,  and  let 
the  rise  be  10.     Then  00  =  21°  48'  =  o.38,/  =  125,  and  k  =  £. 
From  (640), 


From  (74), 


(1.9531  —  i.8i7i)/>=  0.13607'; 

0.1360 

- — — •  =  0.069  =  relative  error. 

272 


APPENDIX  C,  273 

2°.  Assume  a  single  load  acting  at  the  crown  of  the  arch, 
and  let  the  rise  be  25.  Then  00  =  45°  =  0.785,  p  =  50,  and 
*  =  f 

From  (640), 


From  (74), 

H<  =  -^—  {26041  - 

(0.7813  —  o.7724)/)= 

0.0089 

—  £  —  =  o.oi  14  =  relative  error. 

2°a.  The  same  as  2°,  with  k  —  \. 
From  (640), 


From  (74), 


{  18560  -  37}^  =  0.5503^  ; 


(0.5568  -o. 
0.0065 


0.5568 


=  o.oi  1 6  =  relative  error, 


which  is  practically  the  same  relative  error  which  was  found 
in  2°. 

3°.  Assume  a  single  load  acting  at  the  crown  of  the  arch, 
and  let  the  rise  be  50.     Then  0,  =  63°  26'  =  i.n,  p  =  25,  and 


274  A    TREA  TISE   ON  ARCHES. 

From  (640), 

Hl  =  I  •  — —  (0.3125)7'=  O.39O4/3. 
From  (74), 

//"  = — 152O53  —  40}^  =  o.3894/>; 

2003330* 3 

(0.3904-0.3894)^  = 

O.OOI 


0.3904 


=  0.0025  =  relative  error. 


3°0.  The  same  as  3°,  with  k  =  ±. 
From  (64^), 


From  (74), 


(0.2751  —  o.2744)/>=  0.0007/3 

0.0007 

=  0.0025  =  relative  error. 

The  above  results  are  tabulated  below  for  convenience  in 
comparison. 

HINGED  ARCH  WITH  VERTICAL  LOADS. 


Load  at  Crown,  *  =  i. 

Load  at  Quarter-point,  k  =*  J. 

/// 

Values  of  Ht. 

Values  of  ff, 

(64a) 

(74) 

Diff. 

Rel. 

error,  J{. 

(64a) 

(74) 

Diff. 

Rel. 

error,  *. 

O.IO 

0.25 

0.50 

I-953I 
0.7813 
0.3904 

I.8I7I 
0.7724 
0.3894 

0.1360 
0.0089 
O.OOIO 

6.9 

I.I 
O.2 

0.5568 
0.2751 

0.5503 
0.2744 

0.0065 
0.0007 

6.9 
I.I 

0.2 

APPENDIX  C.  2/5 

The  results  in  the  above  table  are  probably  not  correct  in 
the  fourth  decimal  place,  but  for  our  purpose  they  are  suffi- 
ciently exact. 

The  following  conclusions  may  be  drawn  from  the  tabu- 
lated results  : 

i°.  The  position  of  the  load  has  little  or  no  effect  upon  the 
magnitude  of  the  relative  error. 

2°.  The  common  method  in  general  gives  results  which  are 
too  large. 

3°.  To  obtain  results  which  are  not  six  or  seven  per  cent 
too  large,  the  formulas  wJiich  consider  the  influence  of  the  axial 
stress  must  be  employed  for  flat  arches. 

4°.  For  arches  having  a  rise  equal  to  one  fifth  or  more  of  the 
span  the  common  formulas  are  sufficiently  accurate. 

(b)  Horizontal  Loads. 

A  series  of  computations  similar  to  those  made  for  vertical 
loads  indicated  that  for  arches  having  a  rise  of  one  fifth  or 
more  of  the  span  the  common  formulas  can  be  employed. 
For  flat  arches  the  effect  of  the  axial  stress  should  betaken 
into  account,  as  the  results  obtained  by  the  common  formulas 
are  from  six  to  ten  per  cent  too  small  for  arches  having  a  rise 
of  about  one  tenth  the  span. 

ARCH   WITHOUT  HINGES. 
(a)   Vertical  Loads. 

i°.  Assume  a  single  load  on  the  crown  of  the  arch,  and  let 
the  rise  be  10.     Then  00  =  21°  48'  =  0.38,  /  =  125,  and  k  =  £. 
From  (910), 

//l  =  H?(o.o625)/>  =  '2.3437* 

4  ;.*.":?•*•  ".:.  '.;  ?»rn>-  v'T  .:>'..-, 

From  (101), 

H,  =-0.262616.25  —  0.103  }/>  =  i.6i$oP. 


2/6  A    TREA  TISE   ON  ARCHES. 

(2.3437  -  i.6iso)P 


—  —  -  =  0.309  —  relative  error. 

2-3437 


i°a.  The  same  as  i°,  with  k  =  £. 
From  (910), 


=  1.3162^. 

From  (101), 

Hl  =  0.2626(3.51  —  o.o77)/>  =  0.9007/*. 
(1.3162  —  0.9007)^  =  0.41  5  5/>. 


=  o.  3  1  5          =  relative  error 
1.3162 

2°.  Assume  a  single  load  on  the  crown  of  the  arch  and  let 
the  rise  be  25.  Then  0.  =  45°  =  0.785,  /  =  50,  and  k  =  %. 
From  (gia), 

^  =  ^(0.0625)^=0.9375^ 

From  (101), 

HI  =  0.1419(6.25  —  o.o6)/>=  o.8784/> 

(0-9375  —  0.8784)/1  =  0.0591^. 

0.0501 

----  —  =  0.063  =  relative  error. 

2°a.  The  same  as  2°,  with  k  =  \. 
From  (910), 

ff,  =  15(0.035  i)P=o. 


APPENDIX   C. 


277 


From  (101), 


I  —  0.1419(3.51  —  o.04)P  = 
-  0.4910)  = 
0.0355 


0.5265 


=  0.067  =  relative  error. 


3°.  Assume  a  single  load  on  the  crown  of  the  arch  and  let 
the  rise  be  50.     Then  00  =  i.n,/  =  25,  and  k  =  %. 
From  (910), 


,  =  ^^(0.0625)^  =  0.4687^. 


From  (101), 


i  =  0.074(6.25  —  o.O2)P  =  0.4610/1 

(0.4687  —  0.46  1  d)P  =  0.007  7  P. 

0.0077 


0.4687 


=  0.017  =  relative  error. 


Collecting  the  above  results  for  convenience,  we  have  the 
following  table : 

ARCH  WITHOUT  HINGES— VERTICAL  LOADS. 


Load  at  Crown,  k  =  i. 


/// 

Values  of//,. 

Values  of  Hl 

„.> 

doi) 

Diff. 

Relative 
Error,  %. 

„„, 

(JOI) 

Diff. 

Relative 
Error,  %. 

31-5 
6.7 

o  10 
0.25 
0.50 

2.3437 
0-9375 
0.4687 

1.6150 

0.8784 

0.4610 

0.7287 
0.0591 
0.0077 

30.9 
6-3 
1-7 

1.3162 
0.5265 

0.9007 
0.4910 

0.4155 
0.0355 

Load  at  Quarter-point,  *  =  }. 


2/8  A    TREA  TISE   ON  ARCHES. 

From  this  table  the  following  conclusions  may  be  drawn : 

1°.  The  position  of  the  load  has  little  or  no  effect  upon  the 
magnitude  of  the  relative  error. 

2°.  The  common  method  in  general  gives  results  which  are  too 
large. 

3°.  In  arches  which  do  not  have  a  rise  equal  to  at  least  one 
fourth  the  span,  the  effect  of  the  axial  stress  is  too  great  to  be 
neglected.  It  amounts  to  about  thirty  per  cent  for  arches  having 
a  rise  equal  to  one  tenth  their  span. 

(b)  Horizontal  Loads. 

A  series  of  computations  similar  to  those  made  for  vertical 
loads  indicated  that  for  loads  near  the  crown  of  the  arch  the 
effect  of  the  axial  stress  can  be  neglected. 

For  loads  near  the  supports  and  the  quarter-points  the 
effect  of  the  axial  stress  amounts  to  at  least  six  per  cent  for 
arches  having  a  rise  of  one  tenth  their  spans,  but  decreases 
rapidly  as  the  ratio  increases. 

Since  horizontal  loads  are  usually  caused  by  wind,  and  the 
ratio  of  the  wind  stresses  to  the  live  and  dead  load  stresses  is 
small  (ordinarily),  the  common  method  is  probably  sufficiently 
exact  for  practical  purposes. 


CIRCULAR  ARCHES. 

The  above  conclusions  are  based  upon  examples  of  par- 
abolic arches.  For  flat  circular  arches  (rise  less  than  one 
fourth  the  span)  we  can  safely  predict  that  practically  the 
same  conclusions  will  obtain,  since  the  parabola  and  circle  so 
nearly  coincide.  We  will  solve  a  few  examples,  which  will 
show  the  exact  effect  of  the  axial  stress  upon  arches  of 
greater  rise. 


APPENDIX  C.  279 

CIRCULAR  ARCH   WITH    HINGE  AT   EACH  SUPPORT. 

(a)   Vertical  Loads. 
i°.  Let  /=  100  and/  =  25.     Then 

R  =  62.5     and     k'  —  62.5  —  25  =  37.5. 
tan  00  =  ^  =  1.333  ...         .'.  0.  =  53°  71'- 

20n        IO6.25 

—  =  —  5—  ^  =  0.590. 
7T  ISO 

From  Table  XVII,  for  a  =  o,  or  a  load  on  the  crown  : 
For       '  =  0.58,  =  0.758. 


For       °  =  0.60,  =  0.726. 

7T  D 


Then  for          =  0.59,  =  =  0.742. 

7T  02 

From  (160), 


From  (164), 


I  —  —  (sin1  00  —  sin*  a) 

«ir*--5r- 

I   +   g(0o  +  Sin  ^o  COS   0.) 

\vhere  ^  =  0.742^. 


280  A    TREA  TISE   ON  ARCHES. 

From  (153), 

44  4 

m  =  —2 — -  =  — ^,     say  — ^-,     =  0.00102. 
(62.5)°      3906  3900 

From  Table  XVIII. 

for00  =  53°,  6  =  0.1532454 

for  00  =  54°,  B  =  o.  1 67 1 294 


600.0138840 


.00023140    =diff.  for  l' 
71 


0.00179335  =  diff.  for  7!' 
0.1532454 

.%  for  00  =  53°  7f,        B  =  0.1550387 

g  =  0.742.       /.  A  =  (0.742X0.155)  =0.115. 

From  Table  XIX,  by  interpolation, 

(0e  +  sin  00  cos  00)  =  1.407 
Substituting  these  values, 

0.00102,    e 
i (0.64  —  o) 

*,  =  0.742-^? 


O.OOIO2 


or 

^  =  0.733^; 

(0.742  —  o.733)/'  =  0.009^; 

0.009 

=  0.012  =  relative  error; 


APPENDIX  C.  28l 

or  for  a  load  at  the  crown  the  results  by  the  common  method 
formula  (160),  are  about  1.2  per  cent  TOO  LARGE,  which  is 
practically  the  same  as  found  for  parabolic  arches  of  the  same 
rise. 

l°a.  Let  a  load  be  placed  at  the  quarter-point  in  i°.     Then 


...    a  =  23°  35'  =  23^.583,         =  =  0.443. 

Interpolating  in  Table  XVII,  =•  =  0.570. 

D 

From  example  i°, 


B  =  0.155 
.-.     A  =  (0.155X0.570)  =  0.08835. 


From  (160), 


From  c(i\6),  which  is  (164)  in  another  form, 

H  —  p     2A  —  m(s\n*  00  —  sin'  a] 
2B  +  2w(00  -f-  sin  <f>0  cos  00) 


or 

H  =  pai767~  0.00102(0.64-0.16)  = 

0.31287  ' 

(0.570  —  O.563)/5  =  o.oo7/); 

0.007 

-  '  =  0.012  =  relative  error, 

0.570 

which  is  the  same  relative  error  obtained  for  a  load  on  the 
crown. 

2°.  Assume  a  vertical  load  on  the  crown  of  a  semicircular 
arch.    /  =  100,  /  =  50,  and  k  =  £.    Let  (radius  gyration)8  =  4, 

0o  =  90°,  and  a  =  o.     Then  ^  =  i  and  —  =  o. 

71  (t 


282  A    TREATISE   ON  ARCHES. 

From  Table  XVII, 

2d>  a.  A  n 

for  — —  =  i    and    —  =  o,     —  =  0.318. 

7t  (fig  B 

From  (160), 
From  (153), 

Then  from  (164), 


(0.318  —0.317)7'=  o.ooiP; 
=  0.003  =  relative  error, 


0.318 
which  is  too  small  to  be  of  any  practical  importance. 


APPROXIMATE    FORMULAS. 

Very  close  approximate  formulas  for  parabolic  arches  can  be 
formed  by  applying  correction  factors  to  the  common  formulas 
for  vertical  loads. 

Arch  with  a  Pin  at  Each  Support. 

Let  fy  =  the  horizontal  thrust  as  given  by  the  common 
method  ;  then 


where  e  is  the  relative  error. 

Computing  the  value  of  e  for  several  problems  and  plotting 
these  results,  the  following  table  can  be  made  by  means  of  a 
curve  drawn  through  the  plotted  points. 


APPENDIX  C. 


283 


Arch  without  Hinges. 


Results  obtained  by  the  use  of  the  approximate  formulas 
will  be  sufficiently  accurate  for  the  ordinary  problems  met  with 
in  practice. 

VALUES  OF  i  —  e  AND  i  —  e'  IN  THE  APPROXIMATE  FORMULAS  FOR  HI. 


//' 

Hinged. 

Fixed. 

€ 

I  —  € 

f' 

I  -  e* 

O.IO 

.0690 

.9310 

.310 

.690 

O.II 

.0570 

•9430 

.265 

•  735 

0.12 

.0480 

.9520 

.230 

•  770 

0.13 

.O42O 

.9580 

.207 

•  793 

0.14 

.0370 

.9620 

.186 

.814 

0.15 

.0330 

.9670 

.170 

.820 

0.16 

.0295 

•9705 

•153 

.847 

0.17 

.O265 

•  9735 

.140 

.860 

0.18 

.O24O 

.9760 

.126 

.874 

0.19 

.O2I5 

.9785 

.115 

.885 

O.2O 

.0195 

.9805 

.104 

.896 

0.21 

.0170 

.9820 

.094 

.906 

O.22 

•0153 

.9847 

.086 

.914 

0.23 

.0140 

.9850 

.078 

.922 

0.24 

.0125 

.9875 

.070 

•  930 

0.25 

.0115 

.9885 

.065 

•935 

0.26 

.0100 

.9900 

.060 

.940 

0.27 

.0095 

-*  49905 

.058 

.942 

0.28 

.0085 

.9915 

•053 

•947 

0.2Q 

.0080 

.9920 

.050 

.950 

0.30 

.0074 

.9926 

•047 

•953 

0.31 

.0068 

•9932 

•043 

•957 

0.32 

.0063 

•  9937 

.040 

.960 

0.33 

.0060 

.9940 

.038 

.962 

0-34 

.0056 

•  9944 

•035 

.965 

0-35 

.0052 

.9948 

•034 

.966 

0.50 

.0020 

.9980 

.020 

.980 

The  above  table  can  be  used  when  the  arch  rib  does  not  have  too  great  a  varia- 
tion in  6  from  the  crown  to  the  supports.  Increasing  6  at  the  supports'  decreases 
the  effect  of  the  axial  stress.  This  is  quite  marked  in  the  present  form  used  in 
concrete  arches. 


APPENDIX  D. 
SPECIAL  CASE—  SEMICIRCULAR  ARCH. 

2Ed 

—=r-  =  a  constant. 
J\. 

SINCE  semicircular  arches  are  sometimes  employed  for  large 
roof-supports,  we  will  give  the  necessary  formulas  for  determin- 
ing the  outer  forces.  The  effect  of  the  axial  stress  will  be 
omitted,  as  its  effect  can  be  neglected  in  practice.  See  Appendix 
C. 

ARCH   WITH   FIXED   ENDS. 
(a)   Vertical  Loads. 

Tt 

Since  00  =  -,  sin  0,  =  I,  and  cos  00  =  o. 


Then  from  ^(133), 


.       7t      n  . 
2(cos  a  +  oi  sin  a)  ----  sin*  a 


and 


+  cos  a  -f-  «  sin  a 

By  making  a  negative  (hence  the  sin  a  will  be  negative) 
in  the  value  for  M^  we  have 

284 


APPENDIX  D. 


285 


It  Tt  71 

-f  cos  a  -f  a  sm  a  —  _ 

For  any  single  load  we  have,  from  (5 i), 
M. 


Substituting  the  values  of  M^  and  H^   found  above,  we 
obtain  after  reduction 


•      ^ 

>m  a  I  —  — 


cosaJ —  a 


ue 


cos  a  -|-  «  sin  a. 

2  . 


-j~2  cosa-\-2<xs'ma sin*  on 

and  by  making  a  negative  in  the  expression  for^,, 
or  — sin  af—  —  cos  a) 


2R 


From  (50), 


-f-  cos  a  -f-  or  sin  a  -- 


nt     n 
— ( 


Substituting  the  value  of  F,  from  (47)  and  reducing,  this 
becomes 

T  T  PR 

y*  =  -(*  +  sin  «>,  +  -(i  —  sin  a)yt  +  —  cos1  at. 


286  A    TREATISE   ON  ARCHES. 

The  values  of  cos  a  -j-  a  sin  a  can  be  found  from  Table 
XXII. 

(b)  Horizontal  Loads. 
From  £(154), 

±,n  \          — (sin  a  cos  a  —  a)  -j-  2(sin  a  —  a  cos  a) 

TT     ^"(2     I  I       _2 

'  *     1  2         - 

"4 
From  ^(156), 

a  cos  a  —  sin  a  —  I 


=±11^ 

-' 


x      |  -| cos  «  —  cos3  a 

By  making  a  negative  in  the  above  equations,  they  become 
*n   f          -(a  —  sin  a  cos  a)  -f-  2(a  cos  a—  sin  a) 


7? 
2  —  — 


and 


i^(sm«-  M 

it      j  -| cos  a  —  cos*  a  f 


From  (47), 


The  values  of  ^, ,  j, ,  j, ,  and  ^r,  can  be  found  from  (51)  and 
(54> 


APPENDIX  D.  287 

(c)  Effect  of  a  Change  in  Temperature. 
From  £(159), 


From  c(i6i), 


From  (51), 


,     _     Ae? 
~  ~~~  ~          ~' 


M,      2R 
=         =  -=0.632^. 


ARCH  WITH   A  HINGE  AT  EACH   SUPPORT. 

(a)    Vertical  Loads. 
From  <r(io8), 


From 


From  (50), 

V,  i  -f  sin  a       n 


or 

y.  = 


288  A    TREA  TISE   ON  ARCHES. 

(b)  Horizontal  Loads. 
From  c(i2o). 

.   a  —  sin  a  cos  a 


From  <r(i2i), 


From  £(123), 

a  —  sin  a  cos  a 


7T 
2 

The  values  of  a.  —  sin  a  cos  a  can  be  found  from  Table 
XIX. 

(c )  Effect  of  a  Change  in  Temperature. 
From  c(i2S), 


APPENDIX   E. 

DEDUCTION  OF   FORMULAS   FOR  SPECIAL    CASES    FROM 
THE  GENERAL   FORMULAS  OF  CHAPTER  V. 


SYMMETRICAL   PARABOLIC   ARCH. 

(a)  Arch  without  Hinges.     Special  Case  where 
6  cos  (f>  =  a  constant. 

LET  6  cos  0  =  A  =  a  constant,  and  neglect  the  terms  con- 
taining Fx ;  then  for  a  single  vertical  load  we  have  from  ^(90), 
page  117, 

Value  ofH,.— 


—  x)dx 


dx 


P. 


Substituting  the  value  of  y  in  terms   of  x,  the  following 
values  of  the  respective  integrals  are  easily  obtained  :  * 


f\l  - 


=  -//'£(!  ~ 


*  The  general  formulas  for  the  parabola  are  given  on  pages  52  and  53. 


290  A    TREA  TISE   ON  ARCHES. 

(I   -  ty 

-  X}dx  =  \l*k(\  - 


Substituting  these  integrations  in  the  expression  for  //,  and 
reducing, 

_  \fl\k  -  2k*  +  ^)  -  \fl\k  -  V) 
~ 


or 

« 

flr  .........  (90 


i-  —  For  a  single  vertical  load,  from  ^(101),  page 
119,  we  have 


Mt  = 


f'<tx/\U*-(f'x<ix}' 

*/0  t/0  \t/0  / 


Replacing  ^  in  terms  of  x,  the  following  integrations  are 
easily  obtained  : 


-  a)*d*  =  |/'(2  - 


APPENDIX  E. 


Substituting  these  values,  and  that  for  //,  given  above,  we 
obtain 

{!(_  2k  -f  9*  —  I2/P  +  5/£4) 


or 


(92) 


Value  of  H^ — For  a  single  horizontal  load  we  have,  from 
£-(95),  pageiiS, 


We  have  introduced  the  values   of   the   integrals   which 
have  been  determined  above. 

The  values  of  the  two  remaining  integrals  are  as  follows  : 


Hence 


(-  1  5 


1  5) 


Value  of  M^  —  For  a   single  horizontal  load,  from 
page  1  20,  we  have 


292  A    TREA  TISE   ON  ARCHES. 

Performing  the  integrations  indicated, 


s. 
f. 


=  «/'/; 


y  _  fydx  =  -2-(i-6k  +  6kt  +  2k*  - 


Therefore 

Ml  =  Qf\2k(i  —  k}*(2  —  jk  -{-  8/P)}.  .    .    (117) 
The  value  of  Hl  is  given  by  (115). 

(fr)  Arch  with  a  Hinge  at  each  Support. 

As  for  the  arch  without  hinges  let  6  cos  0  =  a  constant, 
and  neglect  the  terms  containing  F# 

Value  of  //",. — For  a  single  vertical  load  we  have,   from 
135, 

(I  -  k)  f'xydx  -  f\x  -  a)ydx 

TT      t/0 t/0 n 

«/0 


The  values  of  the  integrals  are 


-  k] 

O 


,        f'fdx  =  ^/V, 

v/0  X-> 


—  a)ydx  =  —  (i  —  2 


APPENDIX  £. 


293 


Hence 


.    (64) 


Value  of  fft.  —  For  a  single  horizontal  load  we  have,  from 
£•(140),  page  127, 


The  value  of  the  integral  in  the  numerator  of  the  fraction 
is  given  above  in  the  deduction  of  (115);  the  denominator 
equals  -faff;  hence 


.  .    (77) 


SYMMETRICAL  CIRCULAR  ARCH. 


2E 


(a)  Arch  without  Hinges — Special  Case  where  —^-6-=.a  constant. 

Value  of  Ht. — For   a  single  vertical  load  we  have,  from 
£•(90),  page  117,  neglecting  the  terms  containing  Fxy 


-  x)d<t> 


294  A    TREA  TISE   ON  ARCHES. 

Performing  the  integrations  indicated,  we  have,  remember- 
ing that  a  —  klt 


-   f  Xyd<f>  =  -  ^70. 

i/O 

-  (I  -  k)  f 

t/O 


xyd<t>  =  -  #70.  -f  \Kla  +  -  +  ££' 
•       a*       ab.. 


-  k  f'y(l  - 

c/a 


Combining  these  values  we  obtain 

\al  -  i^/0.  +  \k'la  +  bk'  -  ak'a  -     . 


-r 


Combining  these  two  values,  we  have 
aa  —  ^la.  +  f  /00  —  b. 


// 
d<f>     —  200  ; 

.  -  3*7); 


APPENDIX  E.  295 

Substituting  the  values  found  above  for  the  integrals  in- 
dicated in  the  expression  for  //,  it  becomes,  after  reduction, 

_     2bl  -  1(1  -  2a)(&  -a)-  2a>. 
'~ 


which  readily  reduces  to  (192). 

Value  of  M^  —  For   a  single  vertical  load  we   have,  from 
£•(101),  page  119, 


-  pi          ,  n  j  *  nl 

-I  d<t>\-      X*d<t>\-\-      xd<t> 

«/o        I      «/„  )       (      «/o 

The  values  of  the  integrals  not  already  found  above  are 


/ 
*^  =  ^J 


^. 

Then  the  terms  containing  //",  reduce  to 


r  0 

where  d  =  —  ~-. 


296 


A    TREATISE   ON  ARCHES. 


a)d<f>  =  i/0.  4-  */«  -  «00  -  aa  +  £  ; 


Then  for  the  terms  containing  /*we  have 

(l—2d)(b  —  d}<t>t  -\-  2R*a<pQ  -j-  (k1 — d)(2b— 2aa— /00  -\-  la)\. 

The  denominator  becomes  t<pu(d  —  k'}.     Hence 


+  (U  -  d}(2b  -  2aa  -  /0, 


which  reduces  to  (196). 

Value  of  H^. — For  a  single  horizontal  load  we  have,  from 
,  page  1 1 8,  remembering  that  R(—  d<t>)  —  ds, 


.  r^-  (~^W 


I  — 


From  integrals  already  evaluated  the  denominator  of  this 
at  once  becomes 


APPENDIX  E.  297 


2bk'a  ; 


Making  the  proper  substitutions,  we  obtain 

_  r—al-lk'a-lba-ak'fa  -f-  tylfa—abfa—  R^.a—R^* 

~ 


which  reduces  to  (207). 

Value  of  M^  —  For  a  single  horizontal  load  we  have,  from 
£-(106),  page  120, 


1  ~~  /*'      /*'  /    *'      \* 

JfaJ&W  -(/,*<**) 

From  integrals  already  evaluated  the  denominator  becomes 


. 

where 


f\y  -  b}xd<f>  =  -  £(/'  -  «•  -  (^  +  J)K0.  +  «)  +  2*]); 

=  -  (I  -  a  -  ba  -  k'a  -  #0.). 


The  integrals  in  the  terms  containing  ff,  have  been  evaluated 
above. 


298  A    TREATISE  ON  ARCHES. 

Substituting  the  values  determined  above,  we  have 


«)!.    .     .     ,(155) 

which  reduces  to  (212). 

(b]  Arch  with  a  Hinge   at  each  Support  —  Special  Case  where 

2E 

—  6=a  constant. 

J\. 

Value  of  Ht.  —  For  a  single  vertical  load  we  have,  from 
page  125, 


/>; 


Therefore 

Numerator  =  %a(l  —a  —  2k'  a)—  \k\l^  —  la  — 


Hence 


_  D  a(l  -  a  -  2k'  a]  -  k'(l<f>t  -let-  2&) 
*~  -  -• 


which  reduces  to  (160). 


APPENDIX  E.  299 

Value  of  HI.  —  For  a  single  horizontal  load  we  have,  from 
^(140),  page  127, 


=  -  2 


=  —  K*(—  «  +  sin  a  cos  a  +  2  cos  00[sin  or  —  a  cos  a]). 


-      ?*$  =  ^(^o  -  3  c°s  0o  sin  00  +  2  coss  0.0.). 

t/O 

Hence 

Q(          a  —  sin  a  cos  a*  —  2  cos  00(s?n  a  —  a  cos  a  \     /.-2\ 
"i          0;-3cos00sin00  +  2coss0000         f*  V  7  ' 


APPENDIX   F. 

EFFECT  OF  A  COUPLE  UPON  A  SYMMETRICAL  ARCH. 
(a)  Arch  with  a  Hinge  at  each  Support. 

Value  of  H,.  —  Let  Ma  be  any  couple  applied  at  any  point  a 
of  the  arch  :  then 

VJ=M.     or     V,  =  ^. 

Evidently  Vt  is  numerically  equal  to  V^  but  acting  in  the 
opposite  direction,  and  //,  =  Hf 

If  another  couple  equal  and  symmetrical  to  Ma  be  placed 
upon  the  arch, 

ri  =  0=r.    and     fi,  =  2ffl=\  thf  horizontal  thrust  at  the 

|      left  support. 

If  the  arch  be  assumed  free  to  slide  upon  the  supports,  the 
change  in  the  length  of  the  span  due  to  a  horizontal  load  Q 
applied  at  each  support  is  given  by£fn6),  page  122,  or 


Now  let  the  loads  Q  be  removed  and  two  equal  and  sym- 
metrical moments  be  applied  to  the  arch;  the  corresponding 
change  in  the  length  of  the  span  will  be 


where  M,  is  the  resultant  moment  at  any  section  x. 

300 


APPENDIX  F,  301 

If  ^,  represents  the  magnitude  of  the  horizontal  thrust 
necessary  to  cause  a  change  in  the  length  of  the  span  of  the 
loaded  arch  of  A"  I,  we  have 


Substituting  the  values  of  A'  I  and  A"  I,  we  have 


C1MX 

\  -f 

«/0          * 


X  =  O  and  jl/,  =  J/a  from  x  =  a,  to  ^r  = 
Then  since  //,  =  £1^ , 


This  equation  is  perfectly  general  for  any  symmetrical  arch 
having  a  pin  at  each  support. 

(b)  Arch  without  Hinges. 

Value  of  MS — Let  a  couple  Ma  be  applied  at  any  point  a 
on  the  arch;  then  the  moment  at  any  point  x  is 

MM  =  Ml-\-  V,x  -  Hj  +  Ma .  . .  x  >  a. 

Since  the  arch  is  fixed  at  the  ends  J00  =  A<f>t,  and  as  it  is 
symmetrical  Ac  =  o.  Substituting  the  value  of  Mx  in  g(62)  and 
^64),  page  in,  we  obtain,  neglecting  the  axial  stress  term, 


3O2  A    TREATISE   ON  ARCHES. 

and 


Eliminating  Vl  and  solving  for  Mt,  we  have 


in  which  everything  is  known  excepting  Hlt  which  can  be  deter- 
mined as  follows: 

Value  of  H^.  —  Let  two  equal  and  symmetrical  couples  act 
upon  the  arch,  and  assume  the  arch  free  to  slide  upon  the  sup- 
ports.  Also  assume  that  there  are  equal  and  symmetrical 
moments  applied  at  the  supports.  Then  from  ^(62)  we  have 


ClMxds  = 

4/0  * 


Since  the  arch  is  free  to  slide  upon  the  supports,  //,  =  o  ; 
and  since  the  applied  couples  are  equal  and  symmetrical, 
F  =  O.  Therefore 


where  K'  is  the  additional  moment  at  the  section  x  caused  by 
the  action  of  the  applied  couples. 

Substituting  the  value  of  Mx  and  solving  for  Ml 


f'Ei 

J.      <>' 

r±' 

J, «. 


lK'ds 
M,  =  - 


APPENDIX  F.  303 

The  change  in  the  length  of  the  span  due  to  our  couples  is 
(see  ^(80)) 

lK'ds 


Cl 

•/« 


.*• 


Now  suppose  the  arch  unloaded  and  free  to  slide  as  before. 
Let  two  equal  and  symmetrical  moments  Q'z  be  applied  at  the 
supports  ;  then  the  corresponding  change  in  the  length  of  the 
span  is  given  by  ^86),  page  115, 


/x     /•",/      /v 

Q  \     I  y  "s        i  "X  cos  0 

j/=-2rUir+70^--- 


*ds 


Let    ^,   be  the   horizontal   thrust   necessary   to   cause  a 
change  in  the  length  of  span  of  A' I;  then 


or 


- 

C'K'yds         r>Nxdx         0     Bx 
Jo       °*      Vo     F*    '          CldsJ 

/   o* 

__  «^o 

~/     /*/  //  \ 

A/Vc         Cldx  cos  0  _  Wo  ~£j 

Jo  e*J«    F* 


r* 

./o    ^ 


For  x  =  o  to  ^r  =  alt      K'  —  o; 
For  ^r  =  ^  to  x  =  at,     K'  =  Ma ; 
For  ^r  =  at  to  ^r  =  /,,       K'  =  o. 


304  A    TREATISE   ON  ARCHES 

Therefore  since  //,  for  a  single  couple  equals  ffy,  we  have 

/v/i  , 

ids    u.j^  I 


J,    »* 


This  equation  is  perfectly  general  for  any  symmetrical  arch 
which  has  no  hinges. 


PARABOLIC   ARCH. 

0  cos  0  =  a  constant. 

(a)  Arch  with  a  Hinge  at  each  Support. 

Value  of  H^.  —  Our  general  expression  immediately  becomes, 
neglecting  the  axial  stress  term, 


ydx 


Therefore  H,  =  Ma    -  (i  -  6P 


APPENDIX  F.  305 


(J)]  Arch  without  Hinges. 

Value  of  H  '.  —  Neglecting  the  term  which  contains  FXi  we 
have 


/*/.      o 

Jo       dx 


From  Appendix  E,  page  290,  the  denominator  is  found  to 
be^./V. 


r't          / 
^r=rt-(i  -2>&  ; 
v 


-  3*'  + 
Therefore  H,  =  2 


3O6  A    TREA  T1SE   ON  ARCHES. 

Value  of  M^  —  Our  general  equ"  'on  becomes 


_ 
— 


The  denominator  becomes  —  /'. 


fxdx  =  -/2  ;  fl  xydx  —  -Tf. 

t/o  2  i/o     -^  3    J 

Hence 


APPENDIX  G. 
SPECIAL  CASE  WHERE  0  =  A  CONSTANT. 

PARABOLIC  ARCH  WITH  A   HINGE  AT  EACH   SUPPORT— 
VERTICAL   LOAD. 

In  practice  it  sometimes  happens  that  the  arch-rib  has  a 
constant  moment  of  inertia,  especially  in  large  arches.  The 
formulas  already  deduced  do  not  apply  to  such  a  condition, 
though  they  may  be  considered  as  approximately  correct. 

This  case  has  been  very  thoroughly  considered  in  two* 
papers  by  M.  Belliard  in  the  Annales  des  Fonts  et  Chausstes. 
The  principal  results  are  given  below. 

According  to  the  assumption  that  6  cos  0  =  a  constant, 
the  general  equation  for  Hl  becomes, 


(     /»/  (*idx 

pti  _  JA  -j    /    xydx  -|-  6  cos  0  /    -~-  sin  0 

(   i/O  I/O     f1  x 

(     /*f  pidx  ) 

—  P  \    I  y(x  —  «)^ir  -f-  ^  cos  0  /   -^r-  sin  0 1 


#  =  —        -^ 

/y^dx  +  6  cos  0  /    T=T-  cos  0 
i/O    /%. 

while  for  6  =  a  constant 


'f  sin0} 


*  Note  sur  L'erreur  relative  que  Ton  commet  en  substituant  o&r  a  <&  dans  la 
Formule  de  Navier.     April,  1893. 

Memoire  sur  le  calcul  de  la  Resistance  des  arcs  paraboliques  a  grande  fleche. 
November,  1893. 

307 


308  A    TREATISE   ON  AKCHES. 

These  two  equations  are  the  same  in  form,  and  their  only 
difference  is,  in  the  second  ds  replaces  dx  in  the  first  in  all 
terms  excepting  those  containing  Fx. 

Although  the  integration  of  the  second  equation  offers  no 
serious  difficulty,  yet  the  final  results  are  long,  and  their  appli- 
cation in  practice  tedious  without  special  tables. 

The  equation  for  the  common  method  is  very  simple  and 
easy  in  application.  Since  the  location  of  the  load  does  not 
affect  the  relation  between  the  results  obtained  by  using  the 
equation  containing  dx  and  that  containing  d&,  the  relative  error 
between  the  results  can  be  found,  and  the  results  obtained  by 
the  common  method  corrected  to  correspond  with  those  which 
would  have  been  obtained  by  the  application  of  the  correct 
formula  containing  ds. 

M.  Belliard  found  that  the  relative  error  depended  only  upon 

the  ratio  of  the  rise  to  the  span.     For  j  =  0.50  he  found  that 

the  common  formula  H,  =  |  l-?Pk(i  —  k)(i  -f  k  —  /P),  which 
oy 

neglects  the  axial  stress,  gave  a  result  3.3  per  cent  larger  than 
that  given  by  the  exact  formula.  For  ~  =  0.25  the  result 
was  1.7  per  cent  larger,  or  practically  one  half  that  (per  cent) 
for  j  =  50.  This  being  the  case,  it  is  a  very  simple  matter 

to  find  the  percentage  for  any  ratio  of  j  by  interpolation. 

The  magnitudes  of  the  above  errors  are  too  small  to  be  of 
much  practical  importance. 


APPENDIX  H. 

SYMMETRICAL  ARCHES   HAVING   A   VARIABLE   MOMENT 
OF  INERTIA.     SUMMATION   FORMULAS. 

THE  summation  formulas  demonstrated  in  Chapter  V  can 
be  simplified  by  introducing  the  common  moment  for  loads  on 
a  beam  supported  at  the  ends.  The  following  formulas,  while 
approximate,  can  be  safely  applied  in  the  consideration  of  con- 
crete and  reinforced-concrete  arches,  which  usually  have  forms 
which  prevent  the  use  of  the  integration  formulas. 

SYMMETRICAL  ARCH  WITHOUT   HINGES. 

In  g(87),  page  115, 

K'  =  Vlx-I(x-a}  +  IQ(y-l)}>        (x>a,y>b) 
in  which 


Then 

b)  =  mx,       (x>a,y>b) 


which  is  the  common  moment  for  equal  and  symmetrical  loads 
on  a  beam  supported  at  the  ends. 

Neglecting  the  effect  of  the  axial  stress,  £(87)  becomes,  in 
summation  form, 

IvA 


IA 

Js 
in  which  2  =  the  sum  between  the  limits  /  and  o  and  ^  =  z~- 


3IO  A   TREATISE   ON  ARCHES. 

For  Vertical  Loads  only. 


2fft- 

in  which 

mx=R1-2P(x-a),     (x>a) 

the  common  moment  for  equal  and  symmetrical  loads  on  a 
beam  supported  at  the  ends. 

For  horizontal  loads  only,  g(95),  page  118,  becomes 
f 


in  which 

mx=2Q(y-b),     (y>b) 

the  common  moment  for  equal  and  symmetrical  loads  on  a 
curved  beam  supported  at  the  ends. 

The  two  equations  for  Hl  given  above  are  quite  simple  and 
easily  applied.  They  can  be  used  with  equal  facility  for  one 
or  many  loads. 

The  determination  of  Mj  and  M2  will  now  be  considered. 

Ing(7i),  page  112, 

K=-Hiy-P(x-a)  +  Q(y-b).     (x>a,y>b)     g(66) 

This  equation  applies  for  a  single  vertical  and  horizontal  load 
applied  at  the  same  point. 

Let 

b),     (x>a,  y>b) 


APPENDIX  H.  3H 

in  which  R^  is  the  common  reaction  of  P.     Then 
K  =  mx-Rlx-H1y. 

Neglecting  the  effect  of  the  axial  stress,  £(71)  becomes 


l~ 


since  the  arch  is  symmetrical  the  values  of  J  will  be  symmetri- 
cal and  hence  IxA  becomes  -JJ.     For  similar  reasons 

becomes  —2yA.     The  equation  can  now  be  written 


7       Wl\ 

XA[  X  --  ^  —  ~  I 

V       2x4  ) 


readily  reduces  to 


The  value  of  Ml  for  a  load  upon  the  right  of  the  crown  is 
evidently  the  value  of  M2  for  a  corresponding  load  upon  the 
left  of  the  crown.  The  values  of  mx  for  the  load  upon  the 
right  will  be  identical  with  those  for  the  load  upon  the  left 
taken  in  an  inverse  order.  The  value  of  the  expression 

2mxAx—-\  for  the  load  upon  the  right  can  be  found  by 


312 


A    TKEAT1SE   ON  ARCHES. 


using  the  values  of  mx  for  the  load  upon  the  left  and  replacing 
x  by  x'  =  l—  x\   that  is,  instead  of  using  the  values  of  mx  in  an 

inverse  order,  the  values  of  x—  -  are  used  in  an  inverse  order. 
Then  for  a  load  upon  the  right  of  the  crown,  replacing  m^  by  w2> 


ZA 


The  expressions  2 A,  2yA,  and  2A(— jnr]  remain  un- 
changed regardless  of  the  position  of  the  load  considered. 
ImxA  for  a  load  upon  the  right  is  identical  in  value  for  a  cor- 
responding load  upon  the  left.  Therefore 


IA 


in  which 


-b),      (x>a,y>b) 


For  Vertical  Loads  only. 
mx  =  Rix-IP(x-a),     (x>a) 

the  common  moment  for  vertical  loads  on  a  beam  supported 
at  the  ends. 

F,  =the  horizontal  thrust  produced  by  the  loads  considered. 


APPENDIX  H.  313 

For  Horizontal  Loads  only. 


ffa  =  the  horizontal  thrust  at  the  left  support  produced  by 
the  horizontal  loads  considered. 


Effect  of  a  Change  in  Temperature. 

Neglecting  the  effect  of  the  axial  stress,  g(no),  page  121, 
at  once  becomes 


and  g(ii2),  page  122,  reduces  to 


Effect  of  the  Axial  Stress. 

Let  H!  represent  the  horizontal  thrust  at  the  left  support 
produced  by  vertical  loads,  when  the  effect  of  the 
axial  stress  is  neglected,  and 
H a  =  the  horizontal  thrust  at  the  left  support  due  to  the 

axial  stress  corresponding  to  the  vertical  loads. 
An  inspection  of  £(90),  page  117,  shows  that  the  axial  stress 
terms  in  the  numerator  are  comparatively  small  in  effect.  If 
these  terms  be  neglected,  the  numerator  remains  the  same  for 
the  two  cases,  one  when  the  axial  stress  is  considered,  and  the 
other  when  it  is  neglected.  Therefore,  for 


A   TREA  TISE   ON  ARCHES. 


Vertical  Loads  only, 


I— 


2y4(y—£7 


This  may  be  considered  in  effect  equivalent  to  a  drop  in  tem- 
perature which  produces  the  same  horizontal  thrust  and  the 
stresses  in  the  arch  rib  determined  as  if  such  were  actually 
the  condition. 

For  Horizontal  Loads  only. 

The  effect  of  the  axial  stress  produced  by  horizontal  loads 
cannot  be  easily  expressed  independently.  The  horizontal 
thrust  at  the  left  support  when  the  effect  of  the  axial  stress 
is  included  is 


This  expression  differs  from  that  which  neglects  the  effect  of 
the  axial  stress  in  the  denominator  of  the  last  term  only. 


For  Changes  in  Temperature. 


i  — 


In  all  cases 


APPENDIX  H.  315 

This  method  of  considering  the  effect  of  the  axial  stress  is 
approximate,  but  sufficiently  exact  for  practical  purposes. 

Symmetrical  Arch  with  a  Hinge  at  Each  Support. 

In  this  type  of  arch  the  vertical  reactions  are  the  same  as 
for  loads  on  a  beam  supported  at  the  ends.  Ml=M2=o. 
The  expressions  for  Hl  are  readily  determined  and  become  as 
follows: 

For  Vertical  Loads  only. 

From  g(n6),  page  122,  g(ii7),  page  123,  and  the  propor- 
tion at  top  of  page  125, 


in  which 

mx=Rlx-!P(x-a}.    (x>a) 

the  common  moment  for  vertical  loads  on  a  beam  supported 
at  the  ends. 

For  Horizontal  Loads  only. 
From  g(i40),  page  127, 


in  which 


For  Changes  in  Temperature. 
Eefl 


3  I&  A    TREA  TISE   ON  ARCHES. 

Effect  of  the  Axial  Stress. 

Neglecting  the  axial  stress  terms  in  the  numerators  of  the 
general  expressions  for  Hlt  the  above  formulas  can  be  used 

when  the  axial  stress  effect  is  considered  by  adding  2jr  cos  <£ 

f» 

to  ly^A  in  each  formula. 

Note. — A  close  comparison  of  the  above  summation  for- 
mulas with  those  given  on  pages  46  to  50  inclusive  shows  that 
they  are  in  reality  identical.  In  the  new  formulas  mx  has 
been  carried  through  intact,  while  in  the  old  formulas  it  is 
separated  into  two  parts. 

The  new  expression  for  M t  and  M2  is  quite  superior  to  the 
old  form,  as  but  one  half  the  loads  need  be  considered. 

Inasmuch  as  all  summations  are  between  I  and  o,  errors 
introduced  by  using  the  wrong  limits  are  avoided. 


APPENDIX  I. 

UNSYMMETRICAL  ARCHES  WITHOUT  HINGES— SUMMATION 
FORMULAS. 

From  #(59),  g(6o),  and  £(6i),  page  no,  we  have,  if  £  is 
assumed  constant, 

o, (i) 


.       ,  .  ,     .     8s 

in  which  4=-^-. 

Ox 

For  a  single  vertical  load  P  acting  downward  and  a  single 
horizontal  load  Q  acting  toward  the  left  support,  we  have,  from 
(41),  page  16, 


b).        .     (41) 
From  (47),  page  16, 

M2-M,          c       l-a       c-b 

Vi=  — — +HIJ+P~~I — Q~T-     -   •   •    (47) 

Substituting  this  value  of  Fx  in  (41)  it  becomes 

.    .    4     (4) 

317 


318  A   TREATISE  ON   ARCHES. 

in  which 


.     .     (5) 


The  expression  for  mx  is  simply  that  for  the  static  moment 
of  P  and  Q  on  a  curved  beam  which  is  hinged  at  the  right  support 
and  on  rollers  at  the  left  support. 


Combining  (39),  (40),  and  (42),  page  16, 

x>a  x>a 

Nx=H1cos<f)—Qcos<f>  +  ViSm<f>—Psm<f>.    .    .     (6) 
Substituting  the  value  of  Vt  from  (47)  in  (6)  and  letting 

f      I-  a     *><>        c-b}  *><* 

#=|P-j  --  P-Q-pjsin^-^cos^       .     .     (7) 
we  obtain 

N^^^^smt+H^osf+jsm^+K.     .     (8) 

Substituting  the  value  of  M  x  from  (4)  and  the  value  of  Nx 
from  (8)  in  (i),  (2),  and  (3);  eliminating  M2—M^  from  (i) 
and  (2),  (2)  and  (3)  and  then  M^  from  the  two  resulting  expres- 
sions, the  value  of  H  l  is  found  to  be 

ImxB"IA"-ImxA"IE"- 

-  (IK'-IT)  (W'IA"-2B"}-(IK"-ZT'}  VIA" 
1  ~      '""-'""  -        'A"-IB"}  U  +  WU'IA"  ® 


When  the  span  is  divided  in  n  equal  divisions  of  dx  each 


APPENDIX   I. 


319 


and  x=z  —  and  a  =  k  —  the  symbols  in  (9)  represent  the  follow- 
ing expressions: 


2  2 


dx       Fxds 


2 (i5) 


Expressions  (10)  to  (15)  are  independent  of  the  loading  and 
are  constants  for  any  given  arch. 


ZT=et°El,  2T'  =  et°Ec  .....     (16) 


b- 


320  A     TREATISE   ON    ARCHES. 


IT  and  IT'  represent  the  temperature  terms,  and  equations 
(17),  (18),  and  (19)  are  dependent  upon  the  loading. 

If  the  effect  of  the  axial  thrust  is  to  be  neglected  all  terms 
containing  Fx  are  dropped.  If  the  arch  is  symmetrical  then  c 
becomes  zero  and  /  becomes  y. 

The  value  of  M1  can  be  readily  found  from  (i)  and  (3)  after 
MX  and  NX  have  been  substituted.  The  general  form  is 


2 
IG"  IG"    ~        IG"       dx'   ' 


in  which  the  only  new  expression  is 


.         .       (2I) 

[        IzA         Fxds  IzA  \dx/   j 

In  a  similar  manner  the  value  of  M2—  Mx  can  be  found  from 
(i)  and  (3)  and  then  the  value  of  M  2, 


Mt-Mi- 

in  which 


G=J  (z- 


APPENDIX  J. 

UNSYMMETRICAL  ARCH  WITH  TWO   HINGES,  ONE  AT 
EACH   SUPPORT. 


Summation  Formulas. 

There  will  be  no  bending  moments  at  the  supports  and  the 
condition  that  the  central  angle  shall  remain  constant  no  longer 
obtains.  The  length  of  the  span  and  the  relative  positions  of 
the  supports,  however,  must  remain  fixed. 


where 

mx  =  2P^-x-I(x-a)  +  2Q(y-b}-  2QC—j~x.    x>a 

If  the  arch  is  symmetrical,  c  =  o  and 


321 


TABLES. 


TABLES. 

(The  tables  that  follow  are  arranged  according  to  the  scheme  here  given.) 

A.  Tabulated  Properties  of  the  Two-nosed  Catenary. 

B.  A  Series  of  Two-nosed  Catenaries  inscribed  in  the  Circle  of 

Radius  Unity 

B,.  Arch-rings  with  the  Line  of  Stress  lying  within  the  middle 
Third. 

I.      k(l  -  2k* 

8  * 


III.  i  -  |[5(i  -  k  -  2k*  +  4/P) 

IV.  I-2/K2-  5^  +  5^')  +  3^ 
V.     k\     k(i  -  k\     and     (i+k 

VI.     ^(1-^(3-5^)  =  ^. 

VII.       (I  -£)'(!  +2^)=  J7. 

VIII.     ?5£^_2 
5     9^ 

-  2)  _    . 


X.     2/^  - 
XI.    ^a(i  -  ^)2  = 

XII.      I  +  k\-  1  5 

XIII.      2k(l  -  kj(2  -7k  +  8P)  =  4». 

324 


TABLES. 


325 


XIV. 
w 

2.K\\    —  K)  1 

If    —   /K-\~  OK  )                          ^ 

I  +  £'(-15  + 

50*-^  +  ^ 

J\.  V  . 

XVI. 
XVII. 

6k 

1 
-f-  cos  00(cos  a  • 

(sin*  00  —  sin"  a) 
-\-asin  a  —  cos  00  —  00  sin  00) 

200  COS2  < 

^0  —  3  sm  ^o  cos  0o  H~  0o 

XVIII. 
XIX. 

2*°  cos8  *°  —  3  sin  x"  cos  *"  -f-  *~  =  <J.e 
*°  -|-  sin  *°  cos  *°  =  Alt. 
*°  —  sin  *°  cos  *°  =  /#„. 

sin  *°  —  *°  cos  *°  =  ^,,. 

XX. 

^Oi  _|_  x°  sin  ^° 

cos  *°  —  2  sin1  *°  =  JM. 

XXI. 

2  sin  *°  cos  *° 

+  *°  sin1  *°  =  JM. 

XXII. 

cos  *°  -f-  *°  sin  *°  =  An. 

XXIII. 

x°*  _  ^°  sin  x° 

cos  *°  =  J,,. 

XXIV. 

x™  4.  ^°  sin  *° 

cos  *°  =  /J,4. 

XXV. 

Arc  *°     and 

(arc  *°)8. 

XXVI. 

sin  *°,     cos  *°, 

,     and     i  —  cos  *°. 

XXVII. 

sin"  *°,     cos8  * 

°,     sin  *°  cos  *°; 

sin3  *°  and 

cos3  *°. 

XXVIII. 

*°  sin  *°,     *°  i 

_0       _^,     sin  *° 

XXIX. 

*°  sins  *°,     *° 

cos1  *°,     and     *°  sin  *   cos  A 

XXX. 

General     Dimensions    of    Masonry    Arches 

structed  at  Different  Periods. 

XXXI. 

Dimensions  of 

a  few  Cast-iron  Arches. 

XXXII. 

Dimensions  of 

a  few  Wrought-iron  or  Steel  ./ 

XXXIII. 

Dimensions  of 

a  few  Wrought-iron  or  Steel 

truss  Arches. 


326 


A    TREA  TISE   ON  ARCHES. 

TABLE   A.— TABULA    ID    PROPERTIES 


Described  Circle. 

M 

1 

0 

I 

-3 

*T 

•  5774 

O.OOOO 

•  5774 

0.00 

1.7321 

1.7321 

+•5774 

I 

•  32 

— 

.5657 

•2475 

-5831 

8.03 

.7667 

1.7672 

•5657 

I 

•30 

— 

•  5477 

•3977 

.5916 

12.36 

.8187 

1.8224 

•  5477 

c 

.28 

— 

•  5292 

.6000 

15.48 

.8706 

1.8807 

.5290 

1 

.26 

— 

•  5099 

!6ilf 

.6083 

18.21 

.9226 

1.9427 

.5095 

.2 

•  25 

^ 

.5000 

.6585 

.6124 

19.28 

•9485 

1-9754 

.4994 

•o 

•  24 

— 

.4899 

.7041 

.6164 

20.31 

•9745 

2.0092 

.4890 

• 

.22 

— 

.4690 

•  7932 

•6245 

22.24 

2.0265 

2.0810 

.4674 

s 

.20 

— 

•4472 

.8814 

•6325 

24.06 

2.0785 

2.1589 

•  4444 

1 

.19 

— 

•4359 

.9258 

.6364 

24.53 

2.  1044 

2.2007 

.4322 

4> 

.18 

— 

•4243 

.9706 

.6403 

25-37 

2.1304 

2.2446 

.4196 

S3 

•17 

— 

.4123 

1.0163 

.6442 

26.20 

2.1564 

2.2910      .4065 

£ 

.16 

— 

.4000 

1.0630 

.6481 

27.01 

2.1824 

2.3400      .3927 

.0 

•15 

— 

.3873 

I.IIIO 

.6519 

27.40 

2.2084 

2.3922      .3783 

I 

.14 

— 

•3742 

i.  1606 

•6557 

28.18 

2-2344 

2.4477 

•3631 

1 

.13 

— 

.3606 

1.  2122 

.6600 

28.55 

2.26O3 

2.5075 

•3471 

•J3 

.12 

— 

.3464 

1.2663 

.6633 

29.30 

2.2863 

2.5719      .3300 

* 

.1 

§ 

•3333 

I.3I70 

.6667 

30.00 

2.3094 

2.6339]     .3138 

1 

.11 

— 

•3317 

1.3235 

.6671 

30.04 

2.3123 

2.6420      .3117 

.IO 

— 

.3162 

1.3843 

.6708 

30.37 

2.3383 

2.7188 

.2920 

C  3 

.09 

_ 

.3000 

1.4498 

•6745 

31.08 

2.3643 

2.8037 

.2706 

la 

.08 

— 

.2828 

I.52II 

.6782 

31.39 

2.3902 

2.8988 

.2471 

0 

•07 

— 

.2646 

1.5999 

.6819 

32.09 

2.4162 

3.0068      .2209 

0 

.0625 

i 

.2500 

1.6655 

.6847 

32.31 

2.4357 

3.0987      .1990 

1 

.06 

— 

.2450 

1.6888 

.6856 

32.38 

2.4422 

3.1318 

.1912 

•g 

-05 

_ 

.2236 

1.7914 

.6892 

33.06 

2.4682 

3.2801 

.1569 

.04 

^ 

.2000 

1.9141 

.6928 

33-33 

2.4942 

3.4628 

•  "57 

S 

•0357 

— 

.1889 

1-9757 

•6944 

33-45 

2.5053 

3.556i 

.0951 

a 

.03 

— 

•  1732 

2.0688 

.6964 

34.00 

2.5201 

3-6994 

.0639 

i 

.027 

i 

.1667 

2.1096 

•6972 

34.06 

2.5259 

3.7631 

+.0503 

0 

Fora 

value  of 

s  here,  sen 

ibly  the 

next,  dir 

ectrix  tou 

ches  descr 

ibed  circ 

1 

.0204 

* 

.1429 

2.2716 

•6999 

34-25 

2.5451 

4.0191 

-.0037 

1 

•g 

.02 
.01 

A 

.1414 
.1000 

2.2821 
2.6391 

.7000 
•7036 

34.26 
34-51 

2.5461 
2.5721 

4.0360  —.0072 
4.6178:—  .1249 

.005 

— 

.0707 

2.9907 

•7053 

35.04 

2.5851 

5.  2065!  -.2394 

3 

.0048 

— 

.0693 

3.0II4 

•7054 

35-04 

2.5856 

5.2410 

—  .2461 

3 

s 

For  a  va 

ue  of  t 

icre,   sen 

sibly  the  1 

ast,  0a  = 

i 

.OO2 

— 

.0447 

3.4519 

.7064 

35-11 

2.5929 

5.9909 

-.3881 

i 

.OOI 

— 

.0316 

3-7994 

.7068 

35-M 

2-5955 

6.5872 

-•4994 

i 

.OOO 

~ 

.OOOO 

CO 

.7071 

35-16 

2.5981 

00 

—      oo 

For  this  table,  the  modulus  of  common  catenary  from  which  the  members  are  transformed 
two  of  the  above  values  being  given  or  assumed,  the  values  of  the  others  may  be  determined 
for  circular  linear  arches  under  vertical  and  conjugate  horizontal  loads  has  been  calculated 


TABLES. 
OF   THE  "TWO-NOSED   CATENARY." 


327 


Three-point  Circle. 

j-j 

)'i 

t. 

*„ 

/3 

Po  =  Pa 

ft, 

tfj-  /?, 

<o 

ft, 

«. 

o.oooo;  0.5774 

o.oo 

O.OO 

O.OO 

I.732I 

•7321 

.0000 

.OOOO 

.cooo 

.0000 

•351' 

.6009  TO.3O 

11.28 

11.28 

1.7678 

•7673 

.0001 

.OOOO  .OOOO 

.0000 

.5671   .6382;  16.39 

18.08 

18.08 

1.8258 

.8228 

.0003 

.0000  .0000 

.OOOO 

.-'336!  .6780;  — 

22.58 

22.57 

1.8898 

.8818 

.0010 

.0001 

.OCOI 

.0001 

.8808 

.7208 

— 

27.00 

26.56 

1.9612 

•9446 

.0019 

.0004!  .0003 

.0001 

.9507 

•  7435 

26.36 

28.49 

28.44 

2.OOOO 

1-9777 

.0023 

.coo6  .0004 

.0002 

1.0191 

.7671 

— 

30.33 

30.26 

2.O4I2 

2.OI2O 

.0029 

.0009  .0006 

.0004 

I.I533 
1.2884 

.8174 

•  8727 

34.02 

33.48 
36.51 

33.36 
36.33 

2.1320  20846 
2.2361  2.1635 

.0037 

.0046 

.0017  .0013 
.0029  .0022 

.0008 
.0014 

1.3567 

.9025 

— 

38.19 

37-57 

2.2942 

2.2057 

.0050 

.0037 

.0029 

.0018 

1.4260;  .9339 

— 

39.46 

39-20 

2-3570 

2.25OO 

.0054 

.0047 

.0037 

.0023 

1.4969  .9672 

— 

41.11 

40.41 

2.4254  2.2965 

.0055 

.0059 

.0047 

.0030 

1.569(1  1.0026 

— 

42.36 

42.00 

2.5000  2.3456 

.0055 

.0073 

.0059 

.0039 

1.6447  1.0404 

40.24 

44.00 

43  19 

2.5820  2.3975 

.0053 

.0090 

.0074 

.0050 

1.7226 

1.0809 

— 

45-24 

44-37 

2.6726 

2.4527 

.0050 

.OIII 

.0091 

.0063 

1.8040 

1.1247 

— 

46.49 

45-55 

2-7735 

2.5II6 

.0041 

.0135 

.0113 

.0079 

1.8895 

1.1721 

— 

48.14 

47-13 

2.8868  2.5748 

.0029 

.0165 

.0139 

.0100 

1.9697  1.2179 

— 

49-31 

48.22 

3.0000  2.6353 

.0013 

.0195 

.0167 

.0122 

1.9800 

1.2240!  — 

49.41 

48.31 

3.0151  2.6429 

.0009 

.0200 

.0171 

.OI26 

2.0766 

1.2812 

46.13 

51.09 

49-51 

3-1623 

2.7170 

—.0018 

.0242 

.0211 

.OI58 

2.1808 

1-3450 

52.40 

51.12 

3-3333 

2.7982 

-.0055 

.0294 

.0259 

.0199 

2.2944 

1.4170 

— 

54-14 

52.36 

3.5356;  2.8879 

—.0108 

.0358 

.0320  .0253 

2.4202 

1-4997 

— 

55-53 

54-05 

3.7796,  2.9887 

—.0181 

.0437 

.0397  .0324 

2.5247 

1-5709 

— 

57-11 

55-14 

4.0000:  3.0732 

—  .0255 

.0510 

.0470  .0392 

2.5619 

I.5967 

— 

57.38 

55.38 

4-0825 

3.1035 

—  .0283 

.0537 

.0497 

.0418 

2.7255 

1  7139 

51.43 

59-31 

57-20 

4.4721 

3-2374 

—.0427 

.0668 

.0628 

•0549 

2.9210 

1.8613 

61-37 

59.16 

5.0000  3.3985 

-.0642 

.0843 

.0809 

.0738 

3.0189;  1.9382 

— 

62.36 

60.  ii 

5.2926  3.4796 

—.0765 

.0939 

.0911 

.0844 

3.1667  2.0587 

— 

64.01 

61.32 

5  7733  3-6020 

—.0974 

.1093 

•  1073 

.1026 

3.23I5 

2.1131 

— 

64.36 

62.08 

6.0000 

3-6557 

—.1074 

.1164 

.1148 

.1112 

Je;  desc 

ribed  an 

d  three 

-point 

circles 

are  conce 

ntric;  F0 

changes 

sign. 

3.4875:  2.3381 



66.48 

64.22 

7-0000 

3-8678 

-.1512 

.1465 

•  1473 

.1494 

3.5041  2.3532 

— 

66.56 

64.31 

7.07II 

3.8816 

-.1544 

.1486 

.1496 

.1521 

4  0639  2.9107 

— 

7I.O2 

69.20 

10.0000 

4-3432 

—  .2746 

•  2249 

.2337 

.2582 

4  6099  3'5526 

— 

74-  T  7 

7408 

14.1421 

4.7926 

-•4139 

.3101 

.3286 

.3854 

4.6418.  3.5933 

— 

74-27 

74-25 

14.4338 

4.8190 

—  .4220 

.3154 

•3344 

•3924 

ft;  p.  =  >,;•(*, 

-P)  c 

hanges 

sign. 

5-3T9° 

4.565I 



77-39 

80.43 

22.3607 

5.3895 

—  .6013 

.4328 

.4628 

•5659 

5.8495 

5.4874 

— 

79.40 

86.01 

31.6228 

5-8637 

-.7235 

•  5310 

.5654 

.6863 

co 

oo 

57.04 

90.00 

90.00 

OO 

oo 

is-taken  as  unity;  all  quantities  except  s,  r,  and  angles  are  directly  proportional  to  m.  Any 
irom  the  table.  Intermediate  values  can  be  easily  interpolated.  Rankine's  point  of  rupture 
for  certain  values  of  s;  it  is  given  in  the  column  i0. 


328 


A    TREATISE   ON  ARCHES. 
TABLE  B. 


A  Series  of  "  TWO-NOSED  CATENARIES"  inscribed  in  the  Circle  of  Radius  (i?i) 
Unity,  and  having  Parallel  Directrices  at  Graduated  Distances  (Ki  -{-  K0) 
from  its  Centre  from  (l  +  .026)  to  (i  +  .234). 

This  Table  has  for  its  purpose,  in  conjunction  with  supplementary  tables,  the  de- 
signing of  arch-rings,  so  as  to  secure  the  condition  of  the  line  of  stress  lying  within  the 
middle  third,  fifth,  seventh,  etc.,  of  the  arch-ring,  as  may  be  required  to  give  strength  and 
stability  for  every  variation  of  proportion  of  parts  and  of  the  nature  and  distribution  of  load. 


s 

2 

.230 

32  '3 

.4890 

i 

.0016 

•  9783 

0201 

•2345 

.2340 

0007 

.0003 

.225 

21  56 

33  oo 

•4847 

i 

.0017 

.9761 

.0223 

.2299 

.2293 

.0007 

.0003 

.220 

22  24 

33  48 

•  4805 

i 

.0017 

•9739 

•0245 

•  2254 

2247 

.0008 

.0004 

.2IS 

22  50 

34  34 

.4762 

fc  t> 

.0019 

.0*71 

.2208 

.2200 

.0009 

.0004 

23  .6 

35  21 

•4719 

a". 

.0020 

.9687 

.0296 

.2163 

2153 

.0010 

.0005 

.205 

»3  4' 

36  06 

.4675 

w"c 

.0021 

.9658 

.0327 

2117 

2106 

.0011 

.0005 

.195 

2406 
24  30 

3651 
37  35 

.111 

</!  « 

.0022 
.0022 

.9628 
.9596 

.0358 
.0391 

2072 
2026 

2059 

ooi3 

.0007 

:3S 

24  53 

25  15 

38  19 
39  02 

•4544 
•4499 

Ss 

--•  ,c 

.0023 
.0023 

•9563 
•9527 

.0425 
.0463 

.981 
1935 

1964 
1916 

.0017 
.0019 

.0008 
.0009 

.180 

25  37 

39  4<> 

•4455 

-£ 

.0024 

.9491 

.0501 

.,890 

.1869 

.0021 

.0010 

•'75 
.170 
.765 
.160 

2558 

26  20 

26  40 
27  01 

40  28 
41  ii 
41  53 
42  36 

•43'9 
•4273 

This  cir 
die  third, 

.OO24 
.0024 
.0024 
.0024 

•9452 
•94'3 
•9370 
•9327 

•0543 
.0586 
.0635 
.0684 

,1845 
.1800 
•'754 
.1709 

.1821 
•1774 
.1726 
.1678 

.0023 
.OO26 
.0028 
•0031 

.0011 

.0013 
.0017 

•?55 

27  20 

43  18 

.4226 

fS 

.0023 

•9279 

.0738 

.1664 

.1629 

•0034 

.0019 

.150 

27  40 

44  oo 

.4180 

—  ~ 

.0022 

•  9232 

•0793 

.1619 

.0038 

.0031 

•'45 

27  59 

44  42 

•4132 

p.— 

.OO2I 

.9180 

.0856 

•'574 

•1532 

.0042 

.0023 

.140 

28  18 

45  24 

.4085 

cfl*O 

.0020 

.9129 

.0919 

•1529 

.1484 

.0045 

.0026 

•'35 

2836 

4606 

.4036 

C.^ 

.0019 

9071 

.0990 

•1483 

•M34 

.0049 

.0029 

.130 

2855 

46  49 

.3988 

"rt  ** 

.0017 

.9014 

.  06  1 

.1438 

.1384 

.0054 

.0032 

•  125 

29  12 

47  3' 

•3938 

*""•— 

.0014 

.8952 

•  142 

.1392 

1333 

.0059 

•0035 

.120 

29  30 

48  14 

.3888 

•"  jg 

.OOI  I 

.8890 

.  224 

•'347 

.1283 

.0064 

.0039 

.115 

29  47 

48  57 

•3836 

•"•  u 

.0007 

.8821 

•  3l8 

.1301 

.1231 

.0070 

0043 

.110 

3004 

49  4" 

•3785 

U  rt 

.0004 

8752 

412 

1180 

.0076 

.0048 

u^o 

.100 

•095 
.090 

30  37 
30  52 
31  08 

50  25 
Si  09 
5'  54 
52  40 

•1$ 

.3622 
•3567 

11 

'§! 

0.9998 

•9993 
•9987 
.998. 

.8676 
.8600 
•8517 
•8433 

.1521 

iS 

.,889 

.1209 
.1070 

.1127 

1074 

.0082 

.0089 
.0097 

o  05 

.0053 

.085 

31  23 

53  27 

•3508 

5* 

.9972 

.8340 

.2043 

.1023 

!o9ol 

.0  14 

.0079 

.080 

31  39 

54  M 

•  3450 

"  V 

0-5 

.9963 

.8246 

.2197 

.0976 

0852 

o  24 

0087 

•075- 

3i  54 

55  03 

.3388 

So 

•995' 

.8,41 

•2384 

.0928 

•0793 

o  34 

.0097 

.070 
.065 

.060 

32  09 
32  23 
3"  38 

55  53 
56  45 
57  38 

.3326 
•3259 

II 

.9940 
•9925 
.9910 

.8036 

as 

.3036 

.0831 
.0782 

0735 
.0673 
.0611 

:05i 

.0  72 

.0108 
.0131 

0134 

•055 

3»  52 

58  34 

.3121 

11 

•9890 

.7661 

•3335 

•0732 

0544 

0188 

.0150 

.050 

33  06 

59  3' 

.3O49 

.9870 

•  7524 

•3634 

.0682 

'0478 

.0204 

.0167 

.045 

33  20 

to  34 

.2968 

1 

9842 

•7363 

•4037 

.0630 

.0406 

.0223 

.0190 

.040 
•035 

33  33 
3346 

61  37 
62  40 

!38o8 

J 

.9815 
.9788 

.7203 
.7040 

•4440 
4843 

.0578 
.0526 

•  0334 
.0260 

•0243 
.0263 

Independent  of  J?,. 

Directly  proportioned  to  A',,  and  subject  to  any  multiplier. 

For  the  values  of  s  ending  with  5,  on  this  and  on  the  Supplementary  Table,  the  quantities 
are  only  interpolated  as  arithmetical  means,  and  are  correct  to  about  i  per  cent. 


TABLES, 


329 


SUPPLEMENTARY   TABLE   B,. 

Arch  rings  with  the  Two-nosed  Catenary  or  Line  of  Stress  tying  within  the  mid- 
dle third,  and  loaded  from  directrix  to  a  circular  soffit  which  is  the  three-point 
circle  of  another  member  of  the  same  family  of  transformed  catenaries  as  the 
line  of  stress. 


Strong  Brick. 

Average  W't 

Ibs.  p.  cub.  ft. 

Sandstone. 

Average   W't 
140 
Ibs.  p.  cub.  ft. 

Granite. 

Average   W't 

Ibs.  p.  cub.  ft. 

if 

II 
£1<~ 

1 
g 

C     . 

>e 

li 

o  a 

c  3 

1 

A 
1 

Strength 
154,000 

Ibs.  per  sq.  ft. 

Strength 
576,000 

Ibs.  per  sq.  ft. 

Strength 
1,350,000 

Ibs.  per  sq.  ft. 

d  from  ( 
crown  c 

jfl 

JS  « 

54h 

1 

•sf 

SE 

ii 

g 

c 

55*3    • 

11; 

value  of 
m  ic  with 
multiplier. 

^ 

"5 

'1 

c 
1 

s! 
If 

ss 

S..t 

Thickness  of 
to  soffit 

Thickness  of 
to  soffit  at  <f>i 

Radius  of  the 

0 

1 

it 

i 

For  reference. 

1 

3 

98 

|f 

1*1 

1 

1 

~*r 

d 

.. 

ty 

R 

k 

„ 

t 

Feet. 

Feet. 

Feet. 

so 

72 

8.2S 

.141 

.019 

.026 

•973 

•3*5 

•451 

.no 

55 

81 

7-47 

•  137 

.021 

.029 

.970 

•333 

.462 

•  J  J5 

61 

90 

6.70 

•133 

.023 

•°33 

.966 

•34' 

•473 

.110 

68 

101 

6.07 

.129 

.025 

•°37 

.962 

•349 

.482 

.105 

38 

56 

76 

112 

5-45 

.125 

.027 

.041 

•957 

•  49i 

.too 

42 

63 

84 

126 

4-94 

.121 

.029 

•045 

-952 

•  364 

•497 

.095 

46 

7° 

92 

140 

4-44 

.117 

.032 

.049 

.946 

.372 

•  504 

.090 

78 

102 

I56 

4.03 

.113 

.034 

.055 

.940 

.380 

•5°9 

.085 

»9 

39 

57 

86 

"4 

172 

3.62 

.110 

.037 

.062 

•933 

.388 

•5'4 

.080 

ai 

32 

63 

95 

126 

190 

3-31 

.106 

.040 

.071 

•925 

•395 

.516 

.075 

33 

35 

69 

>°5 

138 

3  oo 

.103 

.044 

.081 

.917 

•4°3 

.518 

.070 

SJ 

39 

43 

77 
85 

116 
128 

154 

170 

li 

2.71 
2.42 

.099 
.095 

.048 
.052 

.089 
.097 

.907 
.896 

.409 
.416 

•5'4 

.065 
.060 

31 

47 

94 

'41 

_ 



2.20 

.092 

.056 

10 

.883 

.421 

506 

.055 

104 

'55 



— 

1  .98 

.089 

.061 

•  23 

.869 

.427 

•498 

.050 

42 

I 

~ 

— 

I    fa 

.085 
.082 

.067 

•°73 

41 

•   59 

•85° 

•431 
.436 

.480 
.462 

045 
040 

46 

67 

*-44 

.078 

.078 

•  77 

!si2 

•444 

035 

One  third. 

- 

8 

- 

1 

1 

- 

Directly  prop,  to  tf,.  and  subject  to  any 
multiplier  less  than  given  max. 

- 

'  Note  that  * '+  d  =  \  nearly. 


33° 


A    TREA  TISE   ON  ARCHES. 


TABLE  I. 

VALUES  OF  k(\  -  2/6*  + 


k 

4i 

k 

^i 

k 

A 

k 

4i 

k 

4, 

o 

0 

.21 

0.1934 

.42 

0.3029 

.63 

0.2874 

.84 

0.1525 

.01 

0.0099 

.22 

.2010 

•43 

3052 

.64 

•  2835 

•  85 

.1438 

.02 

.0199 

•23 

.2085 

•44 

•  3071 

•65 

•2793 

.86 

•1349 

•  03 

.0299 

.24 

.2157 

•45 

.3088 

.66 

.2748 

•  87 

•  1259 

.04 

•  0399 

•25 

.2227 

.46 

.3101 

.67 

.2699 

.88 

.1166 

.05 

.0498 

.26 

.2294 

•  47 

.3112 

.68 

.2649 

.89 

.1075 

.06 

.0596 

•27 

•2359 

.48 

•3"9 

.69 

•2597 

.90 

.0981 

.07 

.0693 

.28 

.2422 

•49 

.3124 

•  70 

•  2541 

•9i 

.0886 

.08 

.0790 

.29 

.2483 

•  50 

•  3125 

•  71 

•2483 

•92 

.0790 

.09 

.0886 

•30 

.2541 

•  Si 

.3124 

•  72 

.2422 

•93 

.0693 

.10 

.0981 

•31 

•  2597 

•  52 

•  3119 

•  73 

•2359 

•94 

.0596 

.11 

.1070 

•32 

.2649 

.53 

.3112 

•  74 

.2294 

•95 

.0498 

.12 

.1166 

•  33 

.2699 

•54 

•  3101 

•  75 

.2227 

.96 

•0399 

•13 

•  1259 

•  34 

.2748 

•55 

.3088 

.76 

•  2157 

•97 

.0299 

.14 

•  1349 

•  35 

•  2793 

.56 

•  3071 

•  77 

.2085 

•98 

.0199 

.15 

.1438 

.36 

•  2835 

•  57 

•3052 

•  78 

.2010 

•99 

.0099 

.16 

•  1525 

•37 

•  2874 

•  58 

.3029 

•79 

•1934 

I.OO 

o 

.17 

.1610 

•38 

.2911 

•59 

.3004 

.80 

.1856 

.18 

.1694 

•39 

•  2945 

.60 

.2976 

.81 

.1776 

.19 

.17/6 

.40 

.2976 

.61 

•  2945 

.82 

.1694 

.20 

.1856 

.41 

.3004 

.62 

.2911 

•  83 

.1610 

VALUES 

TABLE  II. 

or8 

5  =  **. 

5  i  +  k  -  i 

k 

J» 

k 

A* 

k 

A* 

k 

//, 

k 

A* 

o 

.6000 

.21 

.3723 

•  42 

.2866 

•63 

1-2975 

.84 

1.4104 

.01 

•5843 

.22 

•3657 

•43 

.2850 

•  64 

1.3004 

•85 

1.4191 

102 

•5692 

•23 

•3593 

•44 

•2837 

.65 

I.3035 

.86 

1.4280 

•03 

•5548 

•24 

•3532 

•45 

.2826 

.66 

1.3068 

•  87 

1-4374 

.04 

.5408 

•25 

•  3474 

.46 

.2816 

.67 

1.3103 

.88 

1.4472 

•05 

.5274 

.26 

.3418 

•47 

.2809 

.68 

1.3141 

.89 

1-4573 

.06 

•5146 

•27 

•  3366 

.48 

.2804 

.69 

1.3181 

.90 

I  4679 

.07 

.5022 

.28 

.33i6 

•49 

.2801 

.70 

1.3223 

.91 

1.4789 

.08 

•4903 

•29 

.3268 

•50 

.2800 

•  7i 

1.3268 

.92 

1.4903 

.09 

.4789 

•30 

.3223 

•51 

.2801 

•  72 

t.33i6 

•93 

1.5022 

,IO 

.4679 

•31 

•  3181 

•52 

.2804 

•  73 

1.3366 

•94 

1.5146 

.11 

•4573 

•32 

•  3141 

•  53 

.2809 

•  74 

1.3418 

•95 

1.5274 

.12 

.4472 

•  33 

•  3103 

•54 

.2816 

.75 

1-3474 

.96 

1.5408 

-13 

•4374 

•34 

.3068 

•  55 

.2826 

.76 

1-3532 

•97 

1.5548 

.14 

.4280 

•35 

•  3035 

•  56 

•  2837 

•  77 

1-3593 

.98 

1.5692 

.15 

.4191 

.36 

.3004 

•  57 

.2850 

.78 

I.3657 

•  99 

1.5843 

.16 

.4104 

-37 

•  2975 

•  58 

.2866 

•  79 

1.3723 

I.OO 

l,6ooo 

-17 

.4022 

•  38 

.2949 

•  59 

.2883 

.80 

1-3793 

.18 

•  3942 

•  39 

.2925 

.60 

•  2903 

.81 

1.3866 

.19 

.3866 

•40 

2903 

.61 

.2925 

.82 

1.3942 

.20 

•3793 

•41 

•  2883 

.62 

.2949 

•  83 

1.4022 

TABLES. 


33* 


TABLE   III. 


VALUES  OF  i  -  H$&  -  *  -  2*2  + 


* 

.00 

I.OOOO 

.21 

-6137 

.42 

•5025 

.63 

.4892 

.84 

.3217 

.01 

•9753 

.22 

.6029 

-43 

•5017 

.64 

.4865 

.85 

,3066 

.02 

.9510 

•23 

.5927 

•44 

.5011 

•65 

.4834 

.86 

.2909 

.03 

.9274 

24 

•  5831 

•45 

.5006 

,66 

•4799 

.87 

•  2745 

.04 

.9043 

•25 

•  5742 

.46 

.5003 

.67 

.4760 

.88 

•  2573 

•05 

.8818 

.26 

.5659 

•47 

.5001 

.68 

.4716 

.89 

•2395 

.06 

.8600 

.27 

-5583 

.48 

5000 

.69 

.4667 

.90 

.2210 

-07 

.8387 

.28 

•  5512 

.49 

,5000 

.70 

.4613 

•9* 

.2017 

.08 

.8182 

.29 

•  5447 

•  50 

,5000 

•7i 

-4553 

.92 

.1818 

.09 

.7983 

•30 

.5387 

•51 

.5000 

•  72 

.4488 

•93 

.1613 

.10 

•  7790 

•31 

•  5333 

•  52 

.5000 

•  73 

.4417 

.94 

.1400 

.11 

.7605 

•32 

-5284 

•  53 

•4999 

•  74 

•4341 

•95 

.1182 

.12 

.7427 

33 

5240 

•  54 

•4997 

-75 

.4258 

.96 

.0957 

•13 

.7255 

•  34 

.5201 

•55 

•4994 

.76 

.4169 

-97 

0726 

,14 

.7091 

•35 

.5166 

.56 

.4989 

•  77 

.4073 

.98 

.0490 

.15 

.6934 

-36 

.5135 

•  57 

•4983 

•  78 

•397r 

•99 

0247 

.16 

.6783 

•  37 

.5108 

,58 

-4975 

-79 

•3863 

I.OO 

.0 

.17 

.6640 

•38 

•  5085 

•59 

.4964 

-80 

•3747 

,18 

.6504 

•39 

.5066 

.60 

4950 

,81 

.3625 

.19 

•6375 

.40 

.5050 

.61 

•  4934 

.82 

•3496 

.20 

.6253 

.41 

•  5036 

.62 

•4915 

•  83 

.3360 

TABLE    IV. 

VALUES  OF  i  —  2^(2  — 


k 

J« 

k 

4 

k 

//4 

k 

A 

k 

4 

.00 

I.OOOO 

,21 

5H2 

•42 

.4365 

>63 

.4211 

,84 

.2626 

.01 

.9610 

.22 

.5045 

•43 

•4365 

.64 

.4179 

.85 

.2498 

.02 

•9239 

•23 

4957 

•44 

.4366 

.65 

•4143 

.86 

.2365 

•03 

.8887 

.24 

.4877 

•45 

.4368 

.66 

.4103 

.8? 

.2227 

.04 

.8554 

•25 

.4805 

.46 

•4370 

.67 

•4059 

.88 

.2084 

•05 

.8238 

.26 

•4739 

-47 

-4372 

.68 

.4011 

.89 

.1936 

.06 

.7939 

•  27 

.4681 

.48 

•4373 

.69 

•3959 

.90 

•1783 

.07 

.7656 

.28 

.4629 

•49 

•4375 

.70 

.3903 

.91 

.1625 

.08 

.7390 

.29 

.4583 

.50 

•4375 

•  71 

•  3842 

.92 

.1463 

.09 

•7139 

•30 

•4543 

•51 

4374 

,72 

•  3777 

•93 

,1296 

.IO 

.6903 

•  31 

.4508 

-52 

•4372 

.73 

,3708 

•94 

.1124 

.11 

.6681 

•  32 

.4478 

•  53 

.4369 

.74 

.3634 

•95 

.0948 

.12 

.6473 

•  33 

•4452 

•  54 

.4365 

•75 

•  3555 

.96 

.0767 

•13 

,6279 

•34 

•  4431 

•  55 

.4358 

.76 

-3471 

-97 

.0581 

.14 

,6097 

>35 

.4413 

.56 

.4349 

•  77 

.3383 

.98 

.0392 

.15 

.5928 

.36 

•4398 

•  57 

.4338 

.78 

.3289 

•99 

.0198 

.16 

•  5770 

•37 

.4387 

-58 

.4324 

•  79 

•3I91 

I.OO 

.0000 

•17 

.5624 

•38 

-4378 

•59 

.4307 

.80 

.3088 

.18 

.5488 

•  39 

•4372 

.60 

.4288 

.81 

.2980 

.19 

•  5363 

.40 

.4368 

.61 

4266 

.82 

.2867 

.20 

.5248 

-  -4i 

.4366 

.62 

.4240 

.83 

.2749 

332 


A    TREATISE   ON  ARCHES. 
TABLE  V. 


k 

& 

*<!-*) 

!  4.  k  -  k* 

i--t 

o 

o 

o 

.0000 

I.OO 

OI 

O.OOOI 

0.0099 

.0099 

•99 

02 

.0004 

.0196 

.0196 

.98 

.03 

.0009 

.0291 

.0291 

•  97 

.04 

.0016 

.0384 

.0384 

.96 

.05 

0025 

.0475 

•0475 

•95 

.06 

,0036 

.0564 

.0564 

•  94 

.07 

.0049 

.0651 

.0651 

•93 

.08 

.0064 

.0736 

.0736 

•  92 

.09 

.0081 

.0819 

.0819 

.91 

.10 

.0100 

.0900 

.0900 

90 

.11 

.OI2I 

.0979 

-0979 

•  89 

.12 

.0144 

.1056 

.1056 

.83 

•13 

.0169 

.1131 

.1131 

.87 

.14 

.0196 

.1204 

1204 

.86 

.15 

.0225 

1275 

.1275 

•85 

.16 

.0256 

1344 

•1344 

.84 

.17 

.0289 

.1411 

1411 

•83 

.18 

.0324 

.1476 

.1476 

.82 

,19 

.0361 

-1539 

•  1539 

.81 

,20 

.0400 

.1600 

.1600 

.80 

,21 

.0441 

.1659 

.1659 

•79 

.22 

.0484 

.1716 

.1716 

•73 

.23 

.0529 

.1771 

.1771 

77 

.24 

.0576 

.1824 

.1824 

.76 

•25 

.0625 

.1875 

.1875 

•75 

.26 

.0676 

.1924 

.1924 

•74 

•27 

.0729 

.1971 

.1971 

•73 

.28 

.0784 

.2016 

.2016 

.72 

,29 

.0841 

.2059 

.2059 

•  71 

.30 

.0900 

2IOO 

.2100 

.70 

•31 

.0961 

.2139 

2139 

.69 

.32 

.1024 

2176 

.2176 

.68 

•33 

.1089 

.2211 

2211 

.67 

•34 

.1156 

.2244 

.2244 

.66 

•35 

.1225 

•  2275 

•2275 

•65 

.36 

.1296 

.2304 

.2304 

.64 

•37 

.1369 

.2331 

•2331 

•63 

•  38 

.1444 

.2356 

.2356 

.62 

•39 

.1521 

.2379 

•2379 

.61 

.40 

.1600 

.2400 

.24OO 

.60 

.41 

.1681 

.2419 

.2419 

•59 

.42 

.1764 

.2436 

.2436 

•58 

•43 

.1849 

.2451 

.2451 

•57 

•  44 

.1936 

.2464 

.2464 

•56 

•45 

.2025 

.2475 

•2475 

•  55 

.46 

.2116 

2484 

.2484 

•54 

•47 

.2209 

.2491 

.249! 

•53 

.48 

.2304 

.2496 

.2496 

•52 

•49 

.2401 

.2499 

.2499 

•51 

.50 

.2500 

.2500 

.250O 

•50 

i  -  k 

(I  -  W 

£  (I  -  k) 

1+*-*' 

k 

TABLES. 


333 


TABLE  VI. 
VALUES  OF  k\\  -  k)  (3 


k 

^. 

k 

Tt 

k 

^. 

k 

^/« 

k 

J, 

.0 

o.oooo 

.21 

0.0679 

.42 

0.0920 

Negative. 

Negative. 

.01 

'.OOO2 

.22 

.0717 

•43 

.0895 

.62 

0.0146 

.82 

0.1331 

.02 

.OOII 

•23 

•0753 

•44 

.0867 

•63 

•  O22O 

•83 

.1346 

•03 

.0024 

.24 

.0787 

•45 

•0835 

.64 

.0294 

.84 

•1354 

.04 

.0043 

•25 

.0820 

.46 

.0799 

.65 

•0369 

.85 

-1354 

•05 

.0065 

•26 

.0850 

•47 

.0761 

.66 

.0444 

86 

.1346 

.06 

.0091 

.27 

.0878 

.48 

,0718 

.67 

•  0518 

.87 

.1328 

.07 

.0120 

-28 

.0903 

-49 

•  0673 

.68 

.0591 

.88 

•  1301 

.08 

0153 

.29 

.0924 

•50 

.0625 

.69 

.0664 

.89 

.1263 

.09 

.0185 

•30 

•0945 

•5i 

•0573 

.70 

•0735 

.90 

.1215 

.10 

.0225 

•31 

.0961 

•52 

.0519 

•7i 

.0804 

9i 

•"55 

.11 

.0263 

•32 

.0974 

-53 

.0462 

•72 

.0871 

.92 

.1083 

.12 

.0304 

•33 

.0985 

•54 

0402 

•73 

•0935 

•93 

.0998 

•13 

•0345 

•34 

0991 

•55 

.0338 

•74 

,0996 

•94 

.0901 

.14 

•0373 

•35 

.0995 

•56 

.0275 

•75 

•1054 

•95 

.0785 

•15 

.0430 

•36 

•0995 

-57 

-0209 

•76 

.1109 

.96 

.0663 

.16 

.0473 

•37 

.0991 

•58 

.0141 

•77 

-1159 

•97 

-0522 

•17 

.0515 

.38 

.0984 

•59 

.0070 

•78 

.1204 

.98 

•0365 

.18 

•0557 

•39 

.0973 

.60 

.0000 

•79 

.1245 

•99 

.0191 

.19 

•0599 

•40 

.0960 

Negative. 

.80 

.1280 

1.  00 

.0000 

.20 

.0640 

.41 

.0942 

.61 

O.OO68 

.81 

.1309 

TABLE  VII. 

VALUES  OF  (i  -  £)*(i  -f  zk)  = 


k 

A-, 

k 

Ji 

k 

<4t 

t 

A, 

k 

^t 

o 

1.  0000 

.21 

0.8862 

.42 

0.6189 

•63 

0.3093 

.84 

0,0686 

.01 

o  9997 

.22 

.8760 

•43 

.6043 

.64 

•  2954 

•  85 

=0607 

.02 

.9988 

•23 

.8656 

•44 

•  5895 

•65 

.2817 

.86 

•0533 

.03 

•9973 

.24 

.8548 

•45 

5747 

.66 

.2681 

.87 

.0463 

.04 

•9953 

•25 

•  8437 

.46 

.5598 

.67 

.2548 

.88 

.0397 

.05 

.9927 

.26 

•8323 

•47 

•5449 

.68 

.2416 

,89 

.0336 

.06 

.9896 

•27 

.8206 

.48 

•5299 

.69 

.2287 

.90 

.0280 

.07 

.9859 

.28 

.8087 

49 

•5'49 

.70 

.2160 

9i 

.0228 

.08 

.9818 

.29 

.7964 

•  50 

.5000 

•7i 

.2035 

.92 

.0181 

.09 

•9771 

•30 

.7840 

5i 

.4850 

,72 

.1912 

-93 

.0140 

.10 

.9720 

•31 

.7712 

•  52 

.4700 

•73 

•  1793 

•94 

^0103 

.11 

.9663 

•32 

.7583 

•53 

•  4550 

•74 

.1676 

•95 

.0072 

.12 

.9602 

•33 

•  7451 

•54 

.4401 

•75 

,1562 

96 

.0046 

•13 

•9536 

•  34 

.7318 

•55 

.4252 

.76 

•  MSI 

•97 

.0026 

.14 

.9466 

•35 

.7182 

.56 

,4104 

•77 

•  1343 

.98 

.0011 

•15 

•9392 

.36 

.7045 

•  57 

•3956 

.78 

.1239 

•99 

.0002 

.16 

•  9313 

•37 

.6906 

•  58 

.3810 

•79 

,1137 

1  OO 

.0000 

•17 

.9231 

•38 

.6765 

•59 

.3664 

.80 

.1040 

.18 

.9144 

•39 

.6623 

.60 

•  3520 

.81 

.0945 

.19 

•  9°54 

.40 

.6480 

.61 

•  3376 

.82 

.0855 

.20 

.8960 

•4i 

•6335 

.62 

•3234 

•83 

.0768 

334 


A    TREA  TISE   ON  ARCHES. 


TABLE  VIII. 


VALUES  OF  - 
5 


=  J, 


k 

7t 

k 

A 

• 

^8 

k 

JB 

k 

4. 

Negative. 

Negative. 

.40 

O 

.61 

0.2295 

.82 

0-3415 

o 

.20 

0.6666 

.41 

0.0162 

.62 

.2365 

•83 

•3454 

.01 

26.OOO 

.21 

.6031 

.42 

.0317 

•63 

•  2434 

.84 

•3492 

.02 

12.666 

.22 

•5454 

•43 

.0465 

.64 

.2500 

•85 

•3529 

•03 

8.222 

•23 

.4927 

•44 

.0606 

•65 

.2564 

.86 

•3566 

.04 

6.000 

•  24 

•4444 

•45 

.0740 

.66 

.2626 

•87 

•  3601 

•05 

4.667 

.25 

.4000 

.46 

.0869 

•67 

.2686 

.88 

•  3636 

.06 

3-773 

.26 

•3589 

•47 

.0992 

.68 

•  2745 

.89 

.3670 

.07 

3-143 

.27 

•  3209 

.48 

•  IIII 

.69 

.2802 

.90 

•3704 

.08 

2.667 

.28 

•2857 

•49 

.1224 

•70 

•2857 

.91 

•3736 

.09 

2.296 

.29 

.2528 

•50 

•  1333 

•71 

.2911 

.92 

.3768 

.10 

2.OOO 

•30 

.2222 

•51 

•  1437 

•72 

.2963 

•93 

•3799 

.11 

1-757 

•31 

•1935 

•52 

•  1538 

•73 

•3014 

•94 

•3830 

.12 

1-555 

•32 

.1666 

•53 

.1633 

•74 

.3063 

•95 

•3859 

•13 

1.384 

•33 

.1414 

•54 

.1728 

•75 

•  3"i 

.96 

•  3889 

.14 

1.238 

•34 

.1176 

•55 

.1818 

.76 

•3158 

•97 

•3917 

•15 

i.  in 

•35 

•  0952 

•56 

.1904 

•77 

.3203 

.98 

•3945 

.16 

I.OOO 

•36 

.0740 

•57 

.1988 

.78 

.3248 

•99 

•3973 

•17 

0.9020 

•37 

.0540 

•58 

.2069 

•79 

.3291 

I.OO 

.4000 

.18 

.8148 

•38 

.0350 

•59 

.2147 

.80 

•3333 

.19 

.7368 

•39 

.OI7O 

.60 

.2222 

.81 

•3374 

1 

TABLE  IX. 


VALUES  OF  10 


k 

^9 

k 

A* 

k 

A» 

k 

4. 

k 

A9 

Negative. 

Negative. 

.40 

o 

.61 

1.4425 

,82 

3-2617 

0 

0 

.20 

0.7143 

.41 

0.0563 

.62 

1.5222 

•83 

3-3595 

.01 

0-0955 

.21 

.7024 

.42 

.1141 

•63 

i  .6029 

.84 

3.4480 

.02 

.1826 

.22 

.6875 

•43 

•1733 

.64 

1.6842 

-85 

3-54I9 

•03 

.2617 

•23 

.6695 

•44 

•  2340 

•65 

1-7663 

.86 

3  6365 

.04 

•3333 

.24 

.6486 

•45 

.2961 

.66 

1.8491 

•  87 

3-7312 

•05 

•3977 

•25 

.6250 

.46 

•3594 

.67 

1.9327 

.88 

3.8266 

.06 

•4553 

.26 

•5987 

•47 

.4240 

.68 

2.0170 

.89 

3-9211 

.07 

•5063 

.27 

•  5698 

.48 

.4898 

.69 

2.IO2O 

.90 

4.0182 

.08 

•5517 

.28 

•5385 

•49 

•5568 

.70 

2.1876 

.91 

4-1148 

.09 

•59" 

•29 

•5047 

•50 

.6262 

•71 

2.2740 

.92 

4-2113 

•  IO 

.6250 

•30 

.4687 

•5i 

•6943 

•72 

2.3608 

•93 

4.3089 

•  II 

•6537 

•31 

•4305 

•52 

.7647 

•73 

2.4484 

•94 

4.4067 

.12 

.6774 

•32 

•  3902 

•53 

•  8361 

•74 

2.5365 

•95 

4-5048 

•13 

.6964 

•33 

•3479 

•54 

.9086 

•75 

2.6250 

.96 

4.6032 

.14 

.7110 

•34 

•  3036 

•55 

.9820 

.76 

2.7144 

•97 

4.7019 

•15 

.7212 

•35 

•2573 

•56 

1.0565 

•77 

2.8044 

.98 

4.8010 

.16 

•7273 

•36 

•  2093 

•  57 

1.1320 

.78 

2.8948 

•99 

4.9008 

•17 

•7295 

•37 

•1594 

•58 

I  .  2083 

•79 

2.9857 

I.OO 

5.0CKX) 

.18 

.7280 

•38 

.1079 

•59 

1.2625 

.80 

3-0771 

.19 

.7229 

•39 

•0547 

.60 

I-3635 

.81 

3-I693 

TABLES. 


335 


TABLE   X. 
VALUES  OF  zk  —  3**  +  /J3  = 


k 

4» 

k 

4* 

k 

^ 

k 

<*», 

k 

4* 

o 

o 

.21 

0.2969 

.42 

0.3848 

•63 

0.3193 

.84 

0.1559 

.01 

0.0197 

.22 

•3054 

•43 

.3848 

.64 

•3133 

•85 

.1466 

.02 

.0388 

•23 

•3134 

•44 

.3843 

.65 

•3071 

.86 

•  1372 

.03 

•0573 

.24 

.3210 

•45 

•3836 

.66 

.3006 

.87 

.1278 

.04 

.0752 

•25 

.3281 

.46 

•3825 

•  67 

.2940 

.88 

.1182 

•05 

.0926 

.26 

•3347 

•47 

.3811 

.68 

.2872 

.89 

.1086 

.06 

.1094 

.27 

•  3409 

.48 

•3793 

.69 

.2802 

.90 

.0990 

.07 

.1256 

.28 

•  3467 

•49 

•  3773 

.70 

.2730 

.91 

.0892 

.08 

.1413 

.29 

•  3520 

•  50 

•  3750 

•  71 

.2656 

.92 

.0794 

.09 

.1564 

•30 

•  3570 

•  5i 

•  3723 

•  72 

.2580 

•93 

.0696 

.10 

.1710 

•31 

.3614 

•  52 

•  3694 

•73 

.2503 

•94 

•0597 

.11 

.1850 

•32 

.3655 

•  53 

.3661 

•74 

.2424 

•95 

.0498 

.12 

.1985 

•33 

.3692 

•  54 

.3626 

•  75 

•2343 

.96 

•0399 

•13 

.2114 

•  34 

.3725 

=  55 

•  3588 

^76 

.2261 

•97 

.0299 

.14 

.2239 

•35 

•3753 

•  56 

.3548 

•77 

.2178 

.98 

.0199 

•15 

.2358 

•36 

•  3778 

57 

•  3504 

.78 

.2093 

•99 

.0099 

.16 

.2472 

•  37 

•3799 

•  58 

•3459 

•79 

.2007 

1.  00 

0 

•17 

.2582 

•  38 

•  3816 

•59 

.3410 

.80 

.1920 

.18 

.2686 

•39 

•  3830 

.60 

.3360 

.81 

.1831 

.19 

.2785 

.40 

.3840 

.61 

.3306 

.82 

.1741 

.20 

.2880 

.41 

.3846 

.62 

3251 

•83 

.1650 

TABLE   XI. 

VALUES  OF 


^2   _   £tj 


k 

#,. 

k 

^., 

k 

<4u 

t 

A. 

i 

A, 

o 

o.oooo 

.21 

0.0275 

.42 

0.0593 

.63 

0.0543 

.84 

0.0180 

.01 

.0000 

.22 

.0294 

.43 

.0600 

.64 

.0530 

•85 

.0162 

.02 

.0003 

•23 

•0313 

•  44 

.0607 

•  65 

•0517 

.86 

.0144 

•03 

.0008 

.24 

.0332 

•45 

.0612 

.66 

.0503 

•  87 

.0127 

.04 

.0014 

•25 

•0351 

.46. 

.0617 

.67 

,0488 

.88 

.OIII 

•05 

.0022 

.26 

.0370 

•47 

.0620 

.68 

•0473 

.89 

.0095 

.06 

.0031 

•27 

.0388 

.48 

.0623 

.69 

0457 

.90 

.0081 

.07 

.0042 

.28 

.O4O6 

•49 

.0624 

.70 

.0441 

.91 

.0067 

.08 

.0054 

•29 

.0423 

•  50 

.0625 

•  7i 

.0423 

.92 

.0054 

.09 

.0067 

•30 

.0441 

•  5i 

.0624 

.72 

.0406 

•93 

.0042 

.10 

.0081 

•31 

•0457 

•  52 

.0623 

•73 

.0388 

•94 

.0031 

.11 

.0095 

•32 

•0473 

•53 

cO62O 

•74 

.0370 

•95 

.0022 

.12 

.0111 

•33 

.0488 

•  54 

.0617 

•75 

•0351 

.96 

.0014 

•13 

.0127 

•34 

.0503 

•  55 

.0612 

.76 

.0332 

•97 

.0008 

.14 

.0144 

•35 

•0517 

•56 

.0607 

•  77 

•0313 

.98 

.0003 

•  15 

.0162 

•36 

.0530 

•57 

.0600 

.78 

.0294 

•99 

.0000 

.16 

.0180 

•37 

•0543 

•  58 

•0593 

•  79 

•0275 

I.OO 

.0000 

•  17 

.0199 

•38 

•0555 

•59 

.0585 

.80 

.0256 

.18 

.0217 

•  39 

.0566 

.60 

.0576 

.81 

.0236 

.19 

.0236 

.40 

.0576 

.61 

.0566 

.82 

.0217 

.20 

.0256 

.41 

•0585 

.62 

•0555 

•83 

.0199 

336 


A    TREATISE   ON  ARCHES. 


TABLE   XII. 

VALUES  OF  i  -f  k\-  15  +  50^  -6o/6»  +  24**)  = 


k 

^n 

k 

J»i 

k 

-rfii 

k 

4U 

k 

J« 

o 

I.OOOO 

.21 

0.6947 

.42 

0.5051 

•63 

0.4790 

.84 

0.2161 

.01 

0.9986 

.22 

.6783 

•43 

•  5034 

.64 

•4739 

•  85 

•  1973 

.02 

•9944 

•23 

.6625 

•44 

.5022 

.65 

.4681 

.86 

.1784 

•03 

.9879 

.24 

•6473 

•45 

•  5013 

.66 

.4616 

•  87 

.1600 

.04 

.9791 

.25 

.6329 

.46 

.5007 

.67 

•  4543 

.88 

.1415 

.05 

.9684 

.26 

.6192 

•47 

.5003 

.68 

.4462 

.89 

•  1234 

.06 

.9561 

•27 

.6063 

.48 

.5002 

.69 

•4374 

.90 

.1058 

•07 

•9493 

.28 

•  5942 

•49 

.5001 

.70 

•4277 

.91 

.0888 

.08 

.9273 

.29 

.5829 

•50 

.5000 

.71 

.4172 

.92 

.0728 

.09 

•9"3 

•30 

•  5724 

•  51 

.5000 

•72 

•4059 

•93 

.0578 

.10 

.8943 

•31 

.5627 

•52 

.5000 

•73 

•3939 

•94 

.0440 

.11 

.8767 

•32 

•  5538 

•  53 

•4999 

•  74 

.3808 

•95 

.0316 

.12 

.8586 

•33 

•  5458 

•  54 

•  4997 

•  75 

.3672 

.96 

.0210 

•13 

.8401 

•  34 

•  5385 

•  55 

.4988 

.76 

.3528 

•97 

.0122 

.14 

.8215 

••35 

•  5320 

•  56 

•  4979 

•77 

•3376 

.98 

.0056 

.15 

.8028 

.36 

.5262 

•  57 

.4967 

.78 

.3218 

•99 

.0015 

.16 

.7841 

•  37 

.5212 

•  58 

•  4951 

•79 

•3054 

I.OO 

0 

•  17 

•  7659 

.38 

•  5i67 

•  59 

.4928 

.80 

.2884 

.18 

.7472 

•  39 

•  5130 

.60 

•4903 

.81 

.2708 

.19 

•7293 

.40 

.5098 

.61 

.4871 

.82 

.2530 

.20 

.7117 

.41 

.5072 

.62 

.4834 

•  83 

.2346 

TABLE  XIII 

VALUES  OF  2k(i  -  k)\z 


k 

J,, 

k 

J,, 

k 

4i, 

k 

4ia 

k 

4ii 

o 

0 

.21 

0.2314 

.42 

0.1331 

•63 

0.1319 

.84 

0.0758 

.01 

0.0073 

.22 

.2268 

•  43 

.13" 

.64 

.1321 

•  85 

.0699 

.02 

.0715 

•23 

.2217 

.44 

.1293 

•  65 

.1321 

.86 

.0639 

•03 

.IOII 

.24 

.2164 

•45 

.1279 

.66 

•1319 

.87 

•0577 

.04 

.1277 

•  25 

.2109 

.46 

.1268 

.67 

.1315 

.88 

•  0515 

.05 

.1506 

.26 

.2052 

.47 

.1260 

.68 

.1307 

.89 

.0453 

.06 

.1705 

•27 

.1994 

.48 

.1254 

.69 

.1298 

.90 

.0392 

.07 

.1874 

.28 

•  1937 

•  49 

.1250 

.70 

.1285 

.91 

•0331 

.08 

.2019 

.29 

.1879 

•  50 

.1250 

•  71 

.1269 

.92 

.0272 

.09 

.2138 

•30 

.1823 

•  51 

.1251 

.72 

.1249 

•93 

.0219 

.10 

.2235 

•31 

.1767 

•  52 

•  1253 

•  73 

.1226 

•  94 

.0168 

.11 

.2311 

•32 

•  I7M 

•53 

.1257 

.74 

.1200 

•95 

.012207 

.12 

.2369 

•33 

.1661 

•54 

.1263 

.75 

.1171 

.96 

.008135 

•13 

.2410 

•34 

.1613 

•55 

.1269 

.76 

."38 

•97 

.004806 

.14 

.2436 

•35 

.1567 

•  56 

.1276 

•  77 

.IIOI 

.98 

.002213 

•15 

.2448 

•36 

.1524 

•  57 

.1283 

•78 

.1062 

•99 

.000576 

.16 

.2448 

•37 

.1483 

•  58 

.1291 

•79 

.1018 

I.OO 

o 

•17 

.2438 

•38 

.1446 

•59 

.1298 

.80 

.0972 

.18 

.2418 

•39 

.1412 

.60 

•  1305 

.81 

.0922 

.19 

.2390 

.40 

.1382 

.61 

.13" 

.82 

.0870 

.20 

.2355 

.41 

•  1355 

.62 

.1316 

•83 

.0814 

TABLES. 


337 


TABLE   XIV. 


VALUES  OF 


-  15  -f  50/6  - 


+  246*) 


i 

^,4 

k 

//,4 

k 

4u 

k 

<*,« 

k 

^.4 

0 

o 

.21 

0-3331 

.42 

0.2635 

•63 

0.2754 

.84 

0.3507 

.01 

0.0073 

.22 

•3344 

•  43 

.2604 

.64 

.2787 

•85 

•  3542 

.02 

.0719 

.23 

.3346 

•44 

•2575 

•65 

.2822 

.86 

•358i 

.03 

.1023 

.24 

•  3343 

•45 

•2551 

.66 

•285-7 

.87 

.3606 

.04 

.1304 

•25 

•  3332 

.46 

•2532 

•  6? 

•2895 

.88 

•  3639 

.05 

.1555 

.26 

•  3314 

•47 

.2518 

.68 

.2929 

.89 

.3670 

.06 

•  1783 

.27 

•  3289 

.48 

•2507 

.69 

.2967 

.90 

•3705 

.0? 

.1976 

.28 

.3260 

•49 

.2501 

.70 

.3004 

.91 

•3731 

.08 

.2177 

.29 

.3224 

•  50 

.2500 

•  71 

.3042 

.92 

•  3748 

.09 

.2346 

•30 

•  3185 

•  51 

.2500 

•  72 

•3077 

•93 

•3797 

.10 

.2499 

•31 

.3140 

•  52 

.2500 

•  73 

•  3"2 

•94 

•3833 

.11 

.2636 

.32 

•  3095 

•  53 

.2514 

•  74 

•3151 

•95 

•3873 

.12 

•2759 

•33 

•  3043 

•  54 

.2527 

•  75 

.3189 

.96 

.3882 

•13 

.2868 

•34 

•  2995 

•  55 

•2544 

•  76 

•3225 

•97 

•  3940 

.14 

.2965 

•  35 

•  2945 

•  56 

.2563 

•77 

.3261 

.98 

•3944 

.15 

•3049 

•36 

.2896 

•  57 

.2583 

•  78 

•  3300 

•99 

.3962 

.16 

.3122 

•37 

.2845 

.58 

.2608 

•79 

•3333 

1.  00 

.4000 

•17 

.3183 

•  38 

•  2793 

•59 

.2633 

.80 

•3370 

.18 

.3236 

•39 

•  2752 

.60 

.2661 

.81 

•3405 

.19 

•3277 

.40 

.2711 

.61 

.2691 

.82 

•3438 

.20 

•3309 

.41 

.2671 

.62 

.2722 

•83 

•347C 

TABLE   XV. 

VWK  <>,*-•>* +  >*'=*,.. 


k 

J,, 

k 

4u 

k 

J,. 

k 

J,« 

k 

4,, 

0 

oo 

.21 

0.7006 

•42 

0.1870 

•63 

0.2024 

.84 

0.3502 

.01 

32.1600 

.22 

.6418 

•43 

.1818 

.64 

•  2075 

•  85 

•  3588 

.02 

15.5266 

•23 

.5892 

•44 

•  1775 

.65 

.2128 

.86 

.3676 

•03 

9.9844 

.24 

.5422 

•45 

.1740 

.66 

.2189 

.87 

.3765 

.04 

7.2116 

•25 

.5000 

.46 

•  1713 

•67 

.2242 

.88 

.3855 

•05 

5.5666 

.26 

.4620 

•47 

.1692 

.68 

.2302 

.89 

•  3945 

.06 

4.4688 

.27 

•4279 

.48 

.1677 

.69 

.2364 

.90 

•4°37 

.07 

3.6888 

.28 

•3947 

•49 

.1669 

.70 

.2428 

.91 

.4126 

.08 

3.1070 

.29 

.3698 

•  50 

.1666 

•  71 

•  2495 

.92 

.4223 

.09 

2.6572 

•30 

•3444 

•  51 

.1669 

•  72 

•  2563 

•93 

.10 

2.3000 

•31 

.3219 

•  52 

.1667 

•73 

.2633 

•  94 

•  4413 

.11 

2.0107 

•32 

.3016 

•53 

.1690 

•  74 

.2705 

•95 

•  4509 

.12 

.7711 

•33 

.2834 

•54 

.1707 

•  75 

•  2778 

.96 

.4606 

•13 

.5709 

•34 

.2670 

•55 

.1727 

.76 

.2853 

•97 

.4703 

.14 

.4011 

•  35 

.2523 

•56 

•  1752 

•  77 

.2929 

.98 

.4801 

•  15 

•  2555 

•36 

.2392 

•57 

.1781 

•  78 

.3007 

•99 

.4900 

.16 

.1301 

•  37 

.2275 

•58 

.1813 

•79 

.3086 

I.OO 

.5000 

•  17 

.0208 

•38 

.2171 

•59 

.1849 

.80 

.3167 

.18 

0.9251 

•39 

.2080 

.60 

.1889 

.81 

•  3249 

.19 

.8410 

.40 

.2000 

.61 

•1931 

.82 

•3332 

.20 

.7666 

.41 

.1930 

.62 

.1976 

•83 

.3416 

338 


A    TREA  TISE   ON  ARCHES. 


TABLE  XVI. 
VALUES  OF  3  —  izk  +  24^ 


=  //,8. 


k 

A. 

k 

AM 

k 

A. 

k 

4n 

k 

4« 

o 

3.0000 

.21 

•  3902 

•42 

.0081 

•63 

0.9648 

.84 

0.3711 

.01 

•  8823 

.22 

•3512 

-•43 

.0054 

.64 

.9560 

•85 

.3140 

.02 

.7694 

•23 

•3M9 

•44 

.0034 

•65 

.9460 

.86 

•2535 

•03 

.6611 

.24 

.2812 

•45 

.0020 

.66 

•9344 

.87 

.1895 

.04 

•5573 

•25 

•  2500 

.46 

.OOIO 

.67 

.9213 

.88 

.1220 

.05 

.4580 

.26 

.2211 

•47 

.0004 

.68 

.9066 

.89 

.0508 

.06 

•3629 

•27 

.1946 

.48 

.OOOI 

.69 

.8902 

Negative*. 

.07 

.2721 

.28 

•1703 

•49 

.OOOO 

.70 

.8720 

.90 

o  .  0240 

.08 

.1854 

.29 

.1481 

•50 

.0000 

•7i 

.8518 

.91 

.IO27 

.09 

.1027 

.20 

.1280 

•5i 

0.9999 

•72 

.8296 

.92 

.1854 

.10 

.0240 

•31 

.1097 

•52 

.9998 

•73 

•8053 

•93 

.2721 

.11 

.9491 

•32 

•0933 

•53 

•9995 

•74 

.7788 

•94 

•3629 

.12 

.8779 

•33 

.0786 

•54 

•9989 

•75 

•7500 

•95 

.4580 

•13 

.8104 

•34 

.0655 

•55 

.9980 

•76 

.7187 

.96 

•5573 

.14 

.7464 

•35 

•0540 

•56 

•99^5 

•77 

.6850 

•97 

.6611 

•15 

.6860 

•36 

•0439 

•57 

•9945 

.78 

.6487 

.98 

.7694 

.16 

.6288 

•37 

•0351 

•58 

.9918 

•79 

.6097 

•99 

.8823 

•17 

•5749 

•38 

.0276 

•59 

•  9883 

.80 

.5680 

1.  00 

I.  0000 

.18 

.5242 

•39 

.0212 

.60 

.9840 

.81 

•5233 

.19 

.4766 

.40 

.OI6O 

.61 

.9787 

.82 

•4757 

.20 

•4320 

.41 

.OIl6 

.62 

•9723 

•83 

-4250 

TABLES. 


TABLE   XVII   (BRESSE). 


VALUES  OF  =-  IN  EQUATION  c(iog)  H^  =  ~SP^-, 
o  a 


339 


2*0 

Value 

•"«• 

o.oo 

0.05 

O.IO 

0.15 

0.2O 

0.25 

0.30 

0-35 

0.40 

0-45 

0.12 

4.125 

4.112 

4-075 

4.012 

3.926 

3.816 

3.682 

3-526 

3-348 

3.I49 

13 

3-804 

3-793 

3-758 

3-700 

3-621 

3.519 

3.396 

3-251 

3.087 

2.903 

.14 

3-529 

3-5i8 

3.486 

3-432 

3-359 

3.264 

3-150 

3.016 

2.863 

2.692 

•15 

3.291 

3.281 

3-251 

3.200 

3-I32 

3-043 

2.936 

2.811 

.669 

2.509 

.16 

3.082 

3.072 

3-044 

2.997 

2-933 

2.862 

2-749 

2.632 

.498 

•349 

•17 

2.897 

2.888 

2.862 

2.817 

2-757 

2.679 

2-584 

•474 

-348 

.207 

.18 

•  733 

2.725 

2.700 

2.657 

2.600 

2.526 

2-437 

•333 

.214 

.081 

.19 

.586 

2.578 

2-554 

•514 

2.460 

•390 

2.305 

.206 

.094 

.968 

.20 

•  453 

•446 

2.423 

-385 

2-334 

.267 

2.187 

•093 

-985 

.866 

.21 

•  333 

.326 

2.304 

.268 

2.219 

.156 

2-079 

•989 

.887 

•774 

.22 

.224 

.217 

2.196 

.162 

2.115 

-054 

1.981 

.895 

•798 

.689 

-23 

.124 

.117 

2.098 

.064 

2.019 

.961 

1.891 

.809 

.716 

.612 

.24 

.032 

.026 

2.007 

•  975 

•932 

.876 

1.809 

-730 

.641 

•  541 

•25 

•947 

.941 

1.923 

-893 

.851 

.798 

1-733 

.658 

•572 

.476 

.26 

.869 

.863 

1.846 

.817 

•777 

.725 

1-663 

-590 

.508 

.416 

.27 

•797 

.791 

1-774 

.746 

.707 

.658 

1.598 

.528 

•448 

.360 

.28 

.729 

.724 

1.708 

.680 

.643 

•595 

1-537 

.470 

•393 

.308 

.29 

.666 

.661 

1.645 

.619 

-583 

•  537 

1.481 

•415 

•341 

-259 

-30 

.607 

.602 

1-587 

.561 

•  527 

.482 

1.428 

.365 

.293 

.213 

•31 

•552 

•  547 

1-533 

.508 

•474 

.431 

1-378 

•317 

.248 

.170 

•32 

.500 

.496 

1.481 

•457 

•424 

.389 

1-332 

.272 

.205 

•  130 

•33 

•452 

•447 

i  433 

.410 

•378 

•337 

1.288 

.230 

.165 

.092 

•34 

.406 

.401 

1-388 

.365 

•  334 

•294 

1.246 

.190 

.127 

1-057 

•35 

.362 

•  358 

1-344 

.322 

.292 

•  254 

1.207 

•153 

.091 

1.023 

•36 

•  321 

•317 

1.304 

.282 

•  253 

.215 

1.170 

.117 

•057 

0.991 

•  37 

.282 

.278 

1.265 

•244 

.216 

.179 

I-I35 

.083 

.025 

.960 

.38 

•  245 

.241 

1.228 

.208 

.180 

.144 

I.IOI 

.051 

0-994 

-931 

-39 

.209 

.205 

1.194 

.174 

.146 

.in 

1.069 

.021 

.965 

•9°4 

.40 

.176 

.172 

1.160 

.142 

.114 

.080 

1.039 

0.991 

•937 

.877 

.42 

•"3 

.109 

1.098 

.080 

•054 

.022 

0.983 

•937 

.885 

.828 

•44 

.056 

.052 

1.042 

.024 

0.999 

0.968 

•931 

.887 

.838 

•783 

.46 

•003 

l.OOO 

0.990 

0.972 

•949 

.919 

.883 

.841 

•794 

•  742 

.48 

0-955 

0.951 

.942 

•925 

•903 

.874 

•  839 

•  799 

•754 

.704 

•  50 

.910 

.907 

.897 

.881 

•  859 

.832 

.798 

.760 

.716 

.668 

•  52 

.868 

.865 

.856 

.840 

.819 

•  793 

.760 

•  723 

.681 

•635 

•  54 

.829 

.826 

.817 

.802 

.782 

•756 

.725 

.689 

.648 

.604 

•  56 

•  793 

.790 

.781 

.767 

•  747 

.722 

.69tr 

•657 

.618 

•  575 

•  58 

•  758 

.756 

•  747 

•  733 

.714 

.690 

.661 

.627 

•  589 

•  548 

.60 

.726 

•723 

•  715 

.702 

.683 

•  659 

.631 

•599 

.562 

.522 

.62 

.696 

.693 

.685 

.672 

•654 

.631 

.603 

•  572 

•  536 

•497 

.64 

.667 

.665 

.657 

.644 

.626 

.607 

•  577 

•  546 

.512 

•474 

.68 

.614 

.612 

.604 

•  592 

•  575 

•  554 

.528 

.499 

.467 

•431 

•  72 

.566 

.564 

•  557 

•  545 

•  529 

.508 

.484 

.456 

.426 

•  392 

.76 

.522 

.520 

.516 

.502 

.486 

.467 

•444 

.417 

.388 

-356 

.80 

.482 

.480 

•473 

.462 

•447 

!  -429 

.406 

.381 

•353 

•323 

.84 

•445 

•443 

.436 

.426 

.411 

•393 

•  372 

•347 

.320 

.292 

.88 

.410 

.408 

.402 

•391 

•  378 

.360 

•  339 

.316 

.290 

.262 

.92 

.378 

•376 

•  370 

•  360 

•  346 

•  329 

•309 

.286 

.261 

•  235 

.96 

•347 

•345 

-349 

-320 

.316 

.300 

.280 

.258 

•  234 

.209 

1.  00 

.318 

•  316 

-3" 

•  301 

.288 

.272 

•  253 

.231 

.208 

.184 

340 


A    TREATISE   ON  ARCHES. 
TABLE   XVII— Continued. 


*. 

w 

Values  of  £-. 

0-50 

0.55 

0.60 

0.65 

0.70 

0-75 

0.80 

0.85 

0.90 

0-95 

O.I2 

2.931 

2.694 

2.441 

2.171 

1.888 

1.592 

1.286 

0.972 

0.651 

0.327 

•13 

2.702 

2.484 

2.250 

2.0OI 

1.740 

1.467 

1.185 

•  895 

.600 

.301 

.14 

2.506 

2.303 

2.086 

I.S55 

i.  612 

1.360 

1.098 

.830 

-556 

•  279 

•  15 

2-335 

2.146 

1-943 

1.728 

1.502 

1.266 

1.023 

•  772 

•  517 

•  259 

.16 

2.186 

2.008 

1.818 

I.6I7 

1.405 

1.184 

0.956 

.722 

.484 

.242 

.17 

2.054 

1.887 

1.708 

1.518 

I-3I9 

1.  112 

.898 

.678 

•454 

.227 

.18 

1.936 

1.778 

1.610 

I-43I 

1.243 

1.048 

•  845 

.638 

.427 

.214 

.19 

1.830 

1.68r 

1.521 

1-352 

1-175 

0.990 

•799 

.603 

•  403 

.202 

.20 

1-735 

1-594 

1.442 

I.28I 

1.  112 

•937 

•  756 

•  571 

-382 

.191 

.21 

1.649 

I.5I4 

1-370 

1.217 

1-057 

.890 

.718 

-542 

.362 

.181 

.22 

I-57I 

1.442 

1.304 

1.  159 

1.006 

.847 

.683 

-SIS 

•344 

.172 

•23 

1.499 

1.376 

1.244 

1.105 

0.959 

.807 

.651 

.491 

.328 

.164 

.24 

1-433 

I.3I5 

1.189 

1.056 

.916 

•771 

.621 

.468 

•313 

•157 

.25 

1-372 

1.259 

1.138 

I.OIO 

.876 

•737 

-594 

.448 

.299 

.149 

.26 

I.3I5 

1.207 

1.091 

0.968 

•839 

.706 

•  569 

.428 

.286 

•MS 

.2? 

1.263 

1.158 

1.047 

.929 

.805 

•677 

•  545 

.411 

.274 

•  137 

.28 

1.214 

1.114 

i.  006 

.892 

-773 

.650 

•  523 

•  394 

.263 

•  131 

.29 

1.169 

1.072 

0.968 

.858 

•  744 

.625 

•503 

•379 

•253 

.126 

•30 

1.126 

1.032 

•  932 

.826 

.716 

.601 

.484 

.364 

•243 

.121 

•31 

1.086 

0.995 

.899 

.796 

.690 

•579 

.466 

•  350 

•234 

.116 

.  -33 

1.049 

.961 

.867 

.768 

.665 

•  558 

•449 

•337 

.225 

.112 

•33 

1.013 

.928 

.837 

•742 

.642 

•539 

•433 

•325 

.217 

.108 

-34 

0.980 

.897 

.809 

.716 

.620 

.520 

.418 

•314 

.209 

.104 

•35 

.948 

.868 

.782 

.693 

•599 

.502 

•403 

•  303 

.202 

.IOO 

.36 

.918 

.840 

•757 

.670 

•579 

.486 

•390 

.292 

.195 

•097 

•37 

.890 

.814 

•733 

.649 

.560 

.470 

•  377 

-283 

.188 

•093 

.38 

.863 

.789 

.711 

.628 

•543 

•454 

•364 

•  273 

.181 

.090 

•39 

•837 

.765 

.689 

.609 

.526 

.440 

•353 

.264 

•175 

.087 

.40 

.812 

•  742 

.668 

.590 

•  509 

.426 

•341 

.256 

.170 

.084 

•42 

.766 

.700 

.629 

.555 

•479 

.400 

.320 

.240 

•  159 

•079 

•44 

.724 

.661 

•594 

•  524 

•451 

•377 

.301 

.225 

.149 

.074 

.46 

.685 

.625 

.561 

•494 

•425 

•345 

.283 

.211 

.140 

.069 

.48 

.650 

•  592 

•  531 

.467 

.401 

•334 

.266 

.198 

.131 

.065 

•  50 

.616 

•  559 

.502 

•442 

•379 

•315 

.251 

.187 

.123 

.O6l 

.52 

.585 

•  532 

.476 

.418 

•  358 

•  297 

.236 

.176 

•  "5 

•057 

.54 

•556 

•  505 

•451 

•396 

•339 

.281 

.223 

.I65 

.108 

•053 

.56 

.529 

.480 

.428 

•375 

.320 

.265 

.210 

•155 

.102 

.050 

•  58 

.503 

•456 

.406 

•  355 

•  303 

.250 

.198 

.146 

.096 

•047 

.60 

•479 

•433 

.385 

.336 

.285 

.236 

.186 

.137 

.090 

.044 

.62 

•456 

.412 

.366 

•319 

.271 

.223 

•  175 

.129 

.084 

.041 

.64 

•434 

•391 

.347 

.302 

.256 

.210 

.165 

.121 

.078 

.038 

.68 

•393 

•354 

•  313 

.271 

.228 

.187 

.146 

.106 

.068 

•033 

.72 

•356 

•  319 

.281 

.242 

.203 

.165 

.128 

.092 

•059 

.028 

.76 

.322 

.287 

.251 

.215 

.180 

•  MS 

.in 

.080 

.050 

.024 

.80 

.291 

.258 

.224 

.191 

.158 

.126 

.096 

.068 

.042 

.019 

.84 

.261 

.230 

.199 

.168 

•137 

.108 

.081 

.057 

•035 

.Ol6 

.88 

•  234 

.204 

•175 

.146 

.118 

.092 

.068 

.046 

.027 

.OI2 

.92 

.208 

.180 

.152 

•  125 

.100 

.076 

.055 

.036 

.021 

.008 

.96 

.183 

•  157 

.131 

.106 

.082 

.061 

.042 

.027 

.014 

.005 

.100 

•  159 

•  134 

.no 

.087 

.066 

•047 

.030 

.017 

.008 

.002 

TABLES. 


341 


TABLE  XVIII. 

VALUES  OF  2x°  cos'2  x°  —  '3  sin  x"  cos  x"  -\-x° 


x° 

4* 

x° 

^..     11  x° 

\ 

J,i 

x" 

4,i 

I 

24 

0.0033256 

47 

0.0870414 

70 

0-5433799 

2 

25 

.0040687 

48 

.0961634 

71 

•5783850 

3 

26 

.0049329 

49 

.1059980 

72 

.6149548 

4 

27 

•0059379 

50 

.1165809 

73 

.6531228 

5 

28 

.0071012 

5i 

.1279485 

74 

.6929174 

6 

29 

.0084352 

52 

.1401378 

75 

.7343691 

7 

30 

.0099597 

53 

.1532454 

76 

.7775070 

8 

31 

.0108680 

54 

.1671294 

77 

.8201831 

9 

32 

.0136523 

55 

.1834638 

78 

.8689473 

10 

33 

.0158626 

56 

.1978582 

79 

.9172998 

ii 

34 

.0183339 

57 

.2153204 

80 

.9674381 

12 

0.0001064 

35 

.0211197 

58 

.2336325 

Si 

1.0193834 

13 

.0001586 

36 

.0242131 

59 

.2508080 

82 

I.073I55I 

14 

.0002295 

37 

.0277103 

60 

.2717593 

83 

1.1287710 

15 

.0003238 

38 

.0314555 

61 

.2930503 

84 

1.18  ? 

16 

.0004463 

39 

.0356564 

62 

•3155469 

85 

i.  25  I 

17 

.0006032 

40 

.0402812 

63 

.339i8o8 

86 

1-30  o 

18 

.0008006 

4i 

.0453581 

64 

.3643046 

87 

1-37  £ 

19 

.0010472 

42 

.0509171 

65 

.3906428 

88 

1-433 

20 

.0013503 

43 

.0569885 

66 

.4183343 

89 

1-50  B 

21 

.0017190 

44 

.0636040 

67 

.4474183 

90 

1.5707963 

22 

.0021640 

45 

.0707964 

68 

•4779307 

23 

.0026954 

46 

.0785976 

69 

.5099063 

342 


A    TREATISE   ON  ARCHES. 


TABLE  XIX. 

VALUES  OF  x°  +  sin  x°  cos  x°  —  J19 
"         "     x"  —  sin  *°  cos  x°  —  fiia 


x° 

4. 

0,, 

X* 

4U 

/3,» 

0 

o 

0 

46 

1.3025470 

0.3031560 

I 

0.0349031 

o.oc  00035 

47 

1.3190868 

.3215226 

2 

.0697848 

.0000284 

48 

.3350190 

.3404970 

3 

.1046243 

.0000955 

49 

.3503454 

.3600772 

4 

.1393998 

.0002266 

50 

.3650685 

.3802607 

5 

.1740906 

.0004424 

5i 

>379I9i7 

.4010441 

6 

.2086737 

.0007659 

52 

.3927190 

.4224234 

7 

•2431339 

.0012121 

53 

•4056552 

.4443938 

8 

.2774450 

.0018076 

54 

.4180060 

.4669496 

9 

.3115881 

.O0257II 

55 

-4297774 

.4900848 

10 

•3455421 

.0035219 

56 

.4409764 

•5137924 

ii 

.3792895 

.0046829 

57 

.4516104 

.5380650 

12 

.4128078 

.O0607I2 

58 

.4616881 

.5628939 

13 

.4460784 

.OO77O72 

59 

.4714926 

.5879960 

14 

.4790819 

.0096103 

60 

.4802103 

.6141849 

15 

.5"7994 

.0117994 

61 

.4886749 

.6406267 

16 

.5442123 

.0142931 

62 

.4966229 

.66758=3 

17 

.5763024 

.0171096 

63 

.5040658 

.6950490 

18 

.6080520 

.O202666 

64 

.5110162 

.7230052 

19 

.6394434 

.0237818 

65 

.5174862 

.7514418 

20 

.6704597 

.0276721 

66 

•5234897 

.7803449 

21 

.7010844 

•0319538 

67 

.5290405 

.8097007 

22 

.7313016 

.0366432 

68 

•534I53I 

.8394947 

23 

.7610956 

.0417558 

69 

•5388425 

.8697119 

24 

.7904514 

.0473066 

70 

.5431243 

.9003367 

25 

•8193545 

.0533101 

7i 

.5470146 

•9313530 

26 

.8477911 

.0597801 

72 

•5505298 

.9627444 

27 

.8757473 

.0667305 

73 

.5536868 

.9944940 

28 

.9032110 

.0741734 

"74 

•5565032 

.0265840 

29 

.9301696 

.0821214 

75 

•5589969 

.0589969 

30 

.9566115 

.0905861 

76 

.5611860 

.0917144 

31 

.9828004 

.0993038 

77 

.5630891 

.1247179 

32 

.0079025 

.1091083 

78 

.5647251 

.1579885 

33 

.0327314 

.1191860 

79 

•5661134 

.1915068 

34 

.0570039 

.1298199 

80 

•5672735 

.2252533 

35 

.0807115 

.1410189 

81 

.5682252 

.2592082 

36 

.1038467 

.1527903 

82 

.5689887 

•2933513 

37 

.1264025 

.1651411 

83 

.5695842 

.3276624 

38 

•1513729 

.1810773 

84 

.5700305 

.3621227 

39 

.1697522 

.1916046 

85 

•5703540 

.3967058 

40 

.1905356 

.2057278 

86 

.5705698 

.4313966 

4i 

.2107191 

.22O45O9 

87 

.5707008 

.4661720 

42 

.2302993 

.2357773 

88 

.5707679 

.5010115 

43 

.2492737 

.2517095 

89 

.5707928 

•5358932 

44 

.2676404 

.2682494 

90 

•5707963 

.5707963 

45 

.2853982 

.2853982 

TABLES. 


343 


TABLE   XIX—  Continued. 
VALUES  OF  sin  x"  —  x"  cos  x°  = 


x° 

//,» 

x° 

Ju 

x° 

Jti 

x° 

^M 

o 

O 

23 

0.0212167 

46 

0.1616323 

69 

O.5O20O59 

I 

24 

.0240716 

47 

.1719073 

70 

.5218362 

2 

25 

.0271669 

48 

•1825754 

71 

.542O8OO 

3 

26 

.0305115 

49 

.1936404 

72 

.5627342 

4 

O.OOOII34 

27 

.0341145 

50 

.2051066 

73 

.5837967 

5 

.OOO2203 

28 

.0379820 

51 

.2169764 

74 

.6052637 

6 

.COO3829 

29 

.0421247 

52 

.2292541 

75 

.6271325 

7 

.OOO6O7I 

30 

.0465502 

53 

.2419419 

76 

.6493983 

8 

.0009047 

3i 

.0512658 

54 

.2550423 

77 

.6720575 

9 

.0012888 

32 

.0562797 

55 

.2685580 

78 

.6951057 

10 

.OCI7668 

33 

.0615995 

56 

.2824910 

79 

.7185380 

ii 

.0023501 

34 

.0672321 

57 

.2968431 

80 

.7423494 

12 

.0030490 

35 

.0731849 

58 

.3116145 

81 

.7665342 

13 

.0038735 

36 

.0794651 

59 

.3268099 

82 

.7910878 

14 

.0048339 

37 

.0860787 

60 

.3424266 

83 

.8160034 

15 

.0059402 

38 

.0930329 

61 

.3584668 

84 

.8412750 

16 

.0072OI9 

39 

.1003340 

62 

.3/49305 

85 

.8668965 

17 

.0086304 

40 

.1079877 

63 

•3919339 

86 

.8928607 

18 

.0103238 

4t 

.1159999 

64 

.4091287 

87 

.9191605 

19 

.OI2O222 

42 

.1243768 

65 

.4268617 

38 

.9457892 

20 

.OI4OO56 

43 

•1331235 

66 

.4450185 

89 

.9927379 

21 

.0161930 

44 

.1422451 

67 

•4635955 

90 

I.OOOOOOO 

22 

.0185936 

45 

.1517464 

68 

.4825916 

344 


A    TREATISE   ON  ARCHES. 


TABLE   XX. 

VALUES  OF  x°*  -\-  x°  sin  x"  cos  x°  —  z  sin5  x° 


jr« 

4« 

x" 

//3«    1 

x° 

Ao 

x° 

A** 

o 

o.oooo 

23 

0.0001818  1 

46 

0.0108522 

69 

0.1100482 

I 

24 

.0002342 

47 

.0122966 

70 

.1191376 

2 

25 

.0002985 

48 

.0138994 

71 

.1290242 

3 

26 

.0003766 

49 

•o:-C577 

72 

.1385871 

4 

27 

.0004688 

50 

.0175990 

73 

.1504997 

5 

28 

.0005851 

5i 

.0197320 

74 

.  16224*42 

6 

29 

.0007206 

52 

.0220699 

75 

.1746965 

7 

30 

.0008810 

53 

.  0246290 

76 

.  1878885 

8 

31 

.0010693 

54 

.0274221 

77 

.1971387 

9 

32 

.0012902 

55 

.0304683 

78 

.2166033 

10 

33 

.0015486 

56 

•0337811 

79 

.2321884 

ii 

34 

.0018452 

57 

•0373797 

80 

.2486336 

12 

35 

.0021893 

58 

.0412871 

81 

.2659695 

13 

36 

.0025843 

59 

.0455069 

82 

•2837735 

14 

37 

.0030365 

60 

.0500725 

83 

.3034412 

15 

38 

•0035517 

61 

.0549993 

84 

.3236408 

16 

39 

.0041368 

62 

.0603087 

85 

.  3448606 

17 

40 

.0047987 

63 

.0663576 

86 

.3671316 

18 

4i 

.0055458 

64 

.0721610 

87 

.3904870 

19 

42 

.0063850 

65 

.0787504 

88 

.4149618 

20 

43 

.0077259 

66 

.0858036 

89 

.4405888 

21 

0.0000776 

44 

.0083771 

67 

.0933561 

90 

.4674012 

22 

0.0001394 

45 

.0095494 

68 

.1014311 

TABLE   XXI. 

VALUES  OF  2  sin  x°  cos  x°  -\-  x°  sin*  x°  = 


x° 

4n 

x° 

J2, 

*• 

An 

x° 

An 

O 

o 

23 

0.7806258 

46 

.4148264 

69 

•7187453 

I 

0.0349049 

24 

.8124424 

47 

.4363274 

70 

.7216027 

2 

.0697989 

25 

.8439761 

48 

•4571858 

71 

.7235986 

3 

.  1046722 

26 

.8752146 

49 

•4773852 

72 

.7244246 

4 

.1385129 

27 

.9061424 

50 

.4959084 

73 

.7243722 

5 

.1743111 

28 

.9367471 

5i 

•5157396 

74 

.7233366 

6 

.2090523 

29 

.9670128 

52 

•5338617 

75 

.7213107 

7 

•2437363 

30 

.9969252 

53 

•5512591 

76 

.7182896 

8 

.2783410 

31 

.0270184 

54 

.5679161 

77 

.7142691 

9 

.3128610 

32 

•0556305 

55 

.5838162 

78 

.7092456 

10 

.3472830 

33 

.0843930 

56 

•5989435 

79 

.7032170 

ii 

.3815964 

34 

.1127420 

57 

.6132827 

80 

.6961813 

12 

.  .4157902 

35 

.1406611 

58 

.6268207 

81 

.6885656 

13 

.4498526 

36 

•1681353 

59 

.  6400866 

82 

.6790869 

14 

•4837723 

37 

•I95I479 

60 

.6514236 

83 

.  6690300 

15 

•5175372 

38 

.2216839 

61 

.6624632 

84 

•6579655 

16 

•5501357 

39 

.2477263 

62 

.6726419 

85 

.6459092 

17 

•5845556 

40 

•2732589 

63 

.6817446 

86 

•6318529 

18 

.6177850 

4i 

.2982657 

64 

.6903668 

87 

.6188062 

19 

.6509107 

42 

.3227295 

65 

.6978876 

88 

•6037755 

20 

.6836206 

43 

.3466341 

66 

.7044952 

89 

•5877332 

21 

.7162018 

44 

•3699635 

67 

.7101816 

90 

.5707963 

22 

.7485413 

45 

•3926991 

68 

•7149359 

TABLES. 


345 


TABLE   XXII. 

VALUES  OF  cos  x°  +  x°  sin  x°  = 


X* 

^*» 

*° 

A,, 

.r0 

A^ 

*" 

A^ 

O 

.OOOO 

23 

•0773544 

46 

1.2721814 

69 

.4826576 

1 

.OOOI52I 

24 

.0839186 

47 

1.2819313 

70 

.4900699 

2 

.0006092 

25 

.0907097 

48 

1.2917062 

71 

.4972383 

3 

.0013697 

26 

.0977206 

49 

1.3014952 

72 

.5042390 

4 

.0024339 

27 

.1049445 

50 

1.3112876 

73 

.5107902 

5 

.0038005 

28 

.1123748 

5i 

1.3210720 

74 

•5I7I505 

6 

.0054680 

29 

.  i  200040 

52 

1.3308372 

75 

•5232129 

7 

.0074354 

30 

.1278248 

55 

1.3405723 

76 

.5289699 

8 

.0097006 

31 

.1358278 

54 

1.3502657 

77 

.5344104 

9 

.0122609 

32 

.1440108 

55 

1.3599061 

78 

•5395194 

10 

.0151152 

33 

.1523602 

56 

1.3694812 

79 

.5442864 

n 

.0182597 

34 

.1608693 

57 

1.3789801 

80 

.5486991 

12 

.0216925 

35 

.1695300 

58 

1.3883923 

81 

•5527461 

13 

.0254098 

36 

•1783332 

59 

1.3977012 

82 

•5564119 

14 

.0294083 

37 

.1872707 

60 

1.4068996 

83 

•5596947 

15 

.0336845 

38 

.1963330 

61 

I.4I59743 

84 

.5625740 

16 

.0382339 

39 

.2055108 

62 

1.4249128 

85 

•5650397 

I? 

.0430530 

40 

.2147947 

63 

1.4337014 

86 

.5670838 

18 

.0481371 

41 

.2241757 

64 

1.4423340 

87 

.56869I3 

!9 

.0534812 

42 

•2336435 

65 

1.4507935 

88 

.5698536 

20 

.0590802 

43 

.2431877 

66 

1.4590655 

89 

•5705585 

21 

1.0649291 

44 

.2527990 

67 

1.4671423 

90 

.5707963 

22 

.0710225 

45 

.2624672 

68 

1.4750112 

TABLE   XXIII 

VALUES  OF  x°'2  —  x°  sin  x°  cos 


* 

An 

*• 

4n 

x° 

4n 

x° 

A» 

0 

23 

0.0167618 

46 

0.2433892 

69 

1.0473746 

I 

24 

.0198158 

47 

.2637466 

70 

1.0998688 

2 

25 

.0232609 

48 

.2852497 

71 

1.1541174 

3 

26 

.0271274 

49 

.3079423 

72 

1.2106700 

4 

27 

.0314472 

50 

.3318402 

73 

1.2670749 

5 

28 

.0362481 

51 

.3569768 

74 

1.3258782 

6 

29 

.0415654 

52 

.3833793 

75 

1.3862235 

7 

O.OOOI48I 

30 

.0564302 

53 

.4110740 

76 

1.4481051 

8 

.0002521 

31 

.0538771 

54 

.4400895 

77 

1.5162235 

9 

.  0004044 

32 

.0609376 

55 

.4704481 

78 

I-576435I 

10 

.0006148 

33 

.0686462 

56 

.5021727 

79 

1.6528610 

ii 

.0008991 

34 

.0770368 

57 

•5352871 

80 

1.7107760 

12 

.0012617 

35 

.0861435 

58 

•5698155 

81 

1.7801637 

13 

.0017487 

36 

.0960009 

59 

.6057683 

82 

1.8514589 

14 

.0023472 

37 

.  1066433 

60 

.6431729 

83 

1.9232828 

15 

.0030891 

38 

.1181055 

61 

.6820439 

84 

1.9969740 

16 

.0039914 

39 

.1304212 

62 

.7223967 

85 

2.0720558 

i? 

.0050765 

40 

.1436249 

63 

.7637878 

86 

2.1485030 

18 

.0063670 

4i 

•1577514 

64 

.8076048 

87 

2.2262890 

19 

.0078863 

42 

.1728338  ' 

65 

.8424868 

88 

2.3053880 

20 

.0096600 

43 

.1885059 

66 

.8988928 

89 

2.3857684 

21 

.0117398 

44 

.2060007 

67 

.9468397 

90 

2.4674012 

22 

.0140700 

45 

.2241512 

68 

•9963335 

346 


A    TREATISE   ON  AKCHES. 


TABLE    XXIV. 

VALUES  OF  x°*  +  x°  sin  x°  cos  x°  = 


X* 

AM 

JT° 

J« 

jr° 

*** 

jr« 

J* 

0 

o 

23 

0.3055234 

46 

1.0457516 

69 

.8531934 

I 

0.0006092 

24 

.3311036 

47 

.0820530 

70 

.8851820 

2 

.0024360 

25 

.3575109 

48 

.1184278 

71 

.9170352 

3 

.0054782 

26 

•3847152 

49 

.1548309 

72 

.9476041 

4 

.0097319 

27 

.4126846 

50 

.1912472 

73 

•9795371 

5 

.0151922 

28 

.4413923 

51 

.2276436 

74 

2.0102920 

6 

.0218524 

29 

.4708012 

52 

.2639917 

75 

2.0407219 

7 

.0297045 

30 

.5008810 

53 

.3002664 

76 

2.0708359 

8 

.0387385 

31 

.5315977 

54 

.3364391 

77 

2.0959325 

9 

.0489436 

32 

.5629190 

55 

.3724885 

78 

2.1301487 

10 

.0603086 

33 

.5948106 

56 

.4083877 

79 

2.1593724 

ii 

.0728185 

34 

.6272388 

57 

.4441165 

80 

2.1883262 

12 

.0864681 

35 

.6601691 

58 

.4796585 

81 

2.2170261 

13 

.1012119 

36 

•6935673 

59 

.5149785 

82 

2.2450355 

14 

.1170628 

37 

.7273991 

60 

.5500725 

83 

2.2737366 

15 

•1339887 

38 

.7616299 

61 

.5849191 

84 

.3017884 

16 

.1519728 

39 

.7962252 

62 

.6195015 

85 

.3296680 

17 

.1709923 

40 

.8311505 

63 

.6531418 

86 

•3573996 

18 

.1911252 

41 

.8663726 

64 

.6878216 

87 

.3850088 

19 

.2120475 

42 

.9018566 

65 

.7215380 

88 

.4125262 

20 

.2339556 

43 

.9379693 

66 

.7549342 

89 

•4399794 

21 

.2569324 

44 

•9734777 

6? 

.7880145 

90 

2.4674012 

22 

.2808096 

45 

1.0095494 

68 

.8207711 

TABLES. 
TABLE   XXV   (WINKLER). 


347 


- 

Arc  x 

(Arc  *)» 

o 

O 

.5707963 

O 

2.4674012 

90 

I 

0.0174533 

.5533430 

0.  0003046 

2.4128739 

89 

2 

.0349066 

•5358897 

.0012185 

2-3589571 

88 

3 

.0523599 

.5184364 

.0027416 

2.3056489 

87 

4 

.0698132 

.5009832 

.0048739 

2.2529513 

86 

5 

.0872665 

.4835299 

.0076154 

2.2008619 

85 

6 

.1047198 

.4660766 

.0109662 

2.1493812 

84 

7 

.1221730 

.4486233 

.0149263 

2.0985097 

83 

8 

.1396263 

.4311700 

.0194953 

2.0482472 

82 

9 

.1570796 

.4137167 

.0246740 

1.9985949 

81 

10 

.1745329 

.3962634 

.0304617 

1.9495511 

80 

ii 

.1919862 

.3788101 

.0368588 

1.9011167 

79 

12 

.2094395 

.3613568 

.0438649 

1.8532919 

78 

13 

.2268928 

•3439035 

.0514803 

I  8060780 

77 

14 

.2443461 

.3264502 

.0597050 

1-7594705 

76 

15 

.2617994 

.3089969 

.0685389 

1.7134727 

75 

16 

.2792527 

.2915436 

.0779821 

1.6680851 

74 

17 

.  2967060 

.2740904 

.0880344 

1.6233060 

73 

18 

.3141593 

.2566371 

.0986961 

1  5791371 

72 

IQ 

.3316126 

.2391838 

.  1099669 

I  5355763 

7i 

20 

.3490659 

.2217305 

.1218476 

1.4925254 

70 

21 

.3665191 

.2042772 

•1343361 

1.4502840 

69 

22 

.3839724 

.1868239 

.1474348 

1.4085523 

68 

23 

.4014257 

.1693706 

.161  1426 

1.3674271 

67 

24 

.4188790 

.1519173 

.1754597 

1.3269135 

66 

25 

•4363323 

.1344640 

.1903859 

1.2870124 

65 

26 

.4537856 

.1170107 

.2059213 

1.2477132 

64 

27 

.4712389 

•0995574 

.2220651 

I  2084648 

63 

28 

.4886922 

.0821041 

.2388202 

1.1709491 

62 

29 

.5061455 

.0646508 

.2561833 

1.1334815 

61 

30 

.5235988 

.0471976 

.2741556 

1.0966227 

60 

31 

.5410521 

.0297443 

.2927374 

1.0603734 

59 

32 

.5585054 

.0122910 

.3119283 

1.0247370 

58 

33 

.5759587 

0.9948377 

.3317284 

0.9897018 

57 

34 

.5934119 

0.9773844 

.3521378 

.9*52802 

56 

35 

.6108652 

0.9599311 

.3731563 

.9214683 

55 

36 

.6283185 

0.9424778 

.3947841 

.8882643 

54 

37 

.6457718 

0.9250245 

.4170212 

.8556702 

53 

38 

.6632251 

0.9075712 

.4398677 

.8236855 

52 

39 

.6806784 

0.8901179 

.4633232 

.7923102 

5i 

40 

.6981317 

0.8726646 

.4873877 

.7615437 

50 

41 

.7155850 

0.8552113 

.5120620 

.7313866 

49 

42 

.7330383 

0.8377580 

.5373452 

.7018386 

48 

43 

.7504916 

0.8203047 

.5632376 

.6728998 

47 

44 

.7679449 

0.8028515 

.5897392 

.6445704 

46 

45 

.7853982 

0.7853982 

.6168503 

.6168503 

45 

Arc  JT 

(Arc  *)* 

JC 

348 


A    TREA  TISE   ON  ARCHES. 
TABLE   XXVI   (WINKLER). 


X 

Sin* 

Cos* 

I—  COS  X 

0 

o 

I 

0 

O 

90 

Z 

0.0174524 

0.9998475 

O.OOOI525 

0.9825476 

89 

2 

.0348995 

.9993910 

.0006090 

.9651005 

88 

3 

•0523359 

.9986293 

.0013707 

.9476641 

87 

4 

.0697565 

.9975640 

.0024360 

.9302435 

86 

%5 

.0871547 

.9961947 

.0038053 

.9128453 

85 

6 

.1045287 

.9945218 

.0054782 

.8954713 

84 

7 

.1218694 

.9925462 

..0074538 

.8781306 

83 

8 

.1391731 

.9902682 

.0097318 

.8608269 

82 

9 

.1564345 

.9876882 

.0123118 

.8435655 

81 

10 

.1736482 

.9848079 

.0151921 

.8263518 

80 

ii 

.1908090 

.9816273 

.0183727 

.8091910 

79 

12 

.2079117 

.9781476 

.0218524 

.7920883 

78 

13 

.2249510 

.9743700 

.0256300 

.7750490 

77 

14 

.2419219 

.9702957 

.0297043 

.7580781 

76 

J5 

.2588190 

.9659258 

.0340742 

.7411810 

75 

16 

.2756374 

.9612614 

.0387386 

.7243626 

74 

17 

.2923717 

.9563046 

.0436954 

.7076283 

73 

-  18 

.3091070 

.9510565 

.0489435 

.6908930 

72 

19 

.3255681 

.9455187 

.0544813 

.6744319 

71 

20 

.3420202 

.9396926 

.0603074 

.6579798 

70 

21 

.3583680 

.9335804 

.0664196 

.6416310 

69 

22 

.3746066 

.9271839 

.0728161 

.6253934 

68 

23 

.39073" 

.9205049 

.0794951 

.6092689 

67 

24 

.4067366 

•9135455 

.0864545 

.5932634 

66 

25 

.4226183 

.9063077 

.0936923 

.5773817 

65 

26 

.4383712. 

.8987941 

.1012059 

.5616288 

64 

2? 

•4539905 

.8910065 

.1089935 

.5460095 

63 

28 

.4694717 

.8829476 

.1170524 

.5305283 

62 

29 

.4848096 

.8746198 

.1253802 

.5151904 

61 

30 

.5000000 

.8660254 

.1339746 

.5000000 

60 

31 

.5150380 

.8571673 

.1428327 

.4849620 

59 

32 

.5299192 

.8480480 

.1519520 

.4700808 

58 

33 

.5446391 

.8386706 

.1613294 

.4553609 

57 

34 

.5591929 

.8290375 

.1709625 

.4408071 

56 

35 

.5735764 

.8191521 

.1808479 

.4264236 

55 

36 

.5877853 

.8090169 

.1909831 

.4122147 

54 

37 

.6018150 

.7986355 

.2013645 

.3981850 

53 

38 

.6156615 

.7880108 

.2119892 

.3843385 

52 

39 

.6293204 

.7771459 

.2228541 

.3706796 

'51 

40 

.6427876 

.  7660446 

•2339554 

.3572124 

50 

41 

.6560589 

.7547096 

.2452904 

.3439411 

49 

42 

.6691306 

.7431449 

.2568551 

.3308694 

48 

43 

.6819983 

.7313537 

.2686463 

.3180017 

47 

44 

.6946584 

.7193398 

.2806602 

.3053416 

46 

45 

.7071068 

.7071068 

.2928932 

.2928932 

45. 

COS  X 

sin  x 

I  —  COS  X 

- 

TABLES 
TABLE  XXVII   (WINKLER). 


349 


- 

sin'* 

cos"  x 

sin  x  cos  x 

sin*  x 

COS'* 

o 

o 

I 

O 

0 

I 

90 

I 

0.0003046 

0.9996953 

0.0174498 

0.0000053 

0.9995428 

89 

2 

.0012180 

.9987822 

.0348782 

.0000425 

.9981739 

88 

3 

.0027391 

.9972609 

.0522644 

.0001437 

.9958942 

87 

4 

.0048660 

.9951340 

.0695866 

.0003394 

.9927100 

86 

5 

.0075961 

.9924037 

.0868241 

.0006620 

.9886273 

85 

6 

.0109262 

.9890738 

.1039539 

.0011421 

•9836552 

84 

7 

.0148521 

.9851477 

.1209609 

.OOlSlOO 

.9778047 

83 

8 

.0193692 

.9806310 

.1378187 

.0026957 

.9710876 

82 

9 

.0244717 

.9755283 

.1545085 

.0038282 

.9635178 

81 

10 

.0301537 

.9698463 

.I7IOIOI 

.005236! 

.9551124 

80 

ii 

.0364080 

.9635920 

•1873033 

.0069470 

.9458883 

79 

12 

.0432273 

.9567727 

.2033683 

.0089875 

.9358650 

78 

13 

.0506030 

.9493969 

.2191856 

.0113832 

.9250638 

77 

14 

.0585262 

.9414737 

.2347358 

.0141588 

.9135079 

76 

15 

.0669873 

.9330127 

.2500000 

.0173376 

.9012213 

75 

16 

.0759760 

.9240239 

.2649596 

.0209418 

.8882288 

74 

17 

.0854812 

.9145187 

.2795964 

.0249923 

•8745788 

73 

18 

.0954915 

.9045085 

.2938927 

.0295085 

.8602386 

72 

19 

.1059947 

.8940055 

.3078308 

.0345085 

.8452989 

71 

20 

.1169778 

.8830222 

.3213938 

.0400088 

.8297694 

70 

21 

.1284274 

.8715726 

•3345653 

.0465573 

.8136828 

69 

22 

.1403301 

.8596700 

.3473292 

.0525686 

.7970722 

68 

23 

.1526708 

.8473292 

.3596699 

.0596532 

•7799707 

67 

24 

.1654347 

.8345653 

•3715724 

.0672884 

•7624135 

66 

25 

.1786062 

.8213938 

.3830222 

.0754823 

•7444355 

65 

26 

.1921693 

.8078308 

.3940055 

.0842415 

.7260735 

64 

27 

.2061079 

.7938921 

.4045084 

.0935708 

.7073636 

63 

28 

.2204036 

•7795964 

.4M5I88 

.1034732 

.6883427 

62 

.29 

.2350403 

.7649599 

.4240241 

,1139498 

.6690480 

61 

30 

.2500000 

.7500000 

.4330127 

.1250000 

.6495189 

60 

31 

.2652642 

.7347358 

.4417483 

.1366211 

.6297915 

59 

32 

.2808144 

.7191857 

.4493971 

.1488090 

.6099040 

58 

33 

.2966310 

.7033684 

.4567727 

.1615573 

.5898943 

57 

34 

.3126968 

.6873033 

.4635920 

.1748579 

.5698003 

56 

35 

.3289899 

.6710101 

.4698463 

.1887009 

.5496592 

55 

36 

•3454915 

.6545085 

.4755282 

.2030748 

.5295083 

54 

37 

.3621813 

.6378187 

.4806307 

.2179661 

.  5093846 

•53 

38 

.3790391 

.6209609 

.4851478 

•2333598 

.4893237 

52 

39 

.3960442 

.6039558 

.4890738 

.2492387 

.4693619 

51 

40 

.4131759 

.5868241 

.4924039 

.2655844 

•4495335 

50 

4i 

.4304134 

.5695866 

•495I34I 

.2823766 

.4298726 

49 

42 

.4477358 

.5522642 

.4972610 

.2995937 

.4104124 

48 

43 

.4651217 

.5348782 

.4987821 

.3172122 

.3911853 

47 

44 

.4825503 

.5174497 

.4996955 

.3352075 

.3722223 

46 

45 

.5000000 

.5000000 

.5000000 

•3535534 

•3535534 

45  ' 

COS5  X 

sin'-r 

sin  x  cos  x 

cos'  x 

sin»  x 

"  x 

350 


A    TREATISE   ON  A  A  CUES. 
TABLE  XXVIII    (WINKLER). 


sin  x 

X 

x  sia  .r 

X  COS  X 

X 

0 

O 

1.5707963 

o 

0 

I 

0.6366197 

90 

I 

0.000304' 

I.553I06I 

0.0174506 

0.0271096 

0.9999486 

.6436748 

89 

2 

.OOI2I82 

1.534954! 

.0348853 

.0536018 

.9997967 

.6506919 

88 

3 

.0027404 

I.5I63554 

.0522881 

.0794688 

.9995428 

.6576697 

87 

4 

.0048699 

1.4973273 

.0696431 

.1047033 

.9991875 

.6646070 

86 

5 

.0076058 

1.4778850 

.0869344 

.1292982 

.9987307   .6715028 

85 

6 

.0109462 

1.4580453 

.1041458 

.1532468 

.9981757!  .6783559 

84 

7 

.0148892 

1.4378255 

.1212623 

.1765428 

.9975147 

.6851651 

83 

8 

.0194324 

1.4172388 

.1382684 

.1991804 

.9967479 

.6919290 

82 

Q 

.0245727 

1.3963116 

.1551457 

.2211540 

.9958928 

.6986465 

81 

IO 

.0303073 

I.3750509 

.1718814 

.2424585 

•9949309 

.7053168 

80 

ii 

.0366327 

1-3534774 

.1884589 

.2630893 

.9938682 

.7H9380 

79 

12 

.0435449 

1.3316077 

.2048627 

.2830419 

.9927055 

.7185100 

78 

J3 

.0510398 

1.3094594 

.2210775 

.3023125 

.9914420 

.7250297 

77 

14 

.0591126 

1.2870480 

.2370880 

.3208974 

.9900789 

.7314980 

76 

15 

.0677587 

1.2643939 

.2528788 

.3387933 

.9886157 

•7379I3I 

75 

16 

.0769725 

1.2415131 

.2684355 

•3559977 

.9870534 

•7442735 

74 

17 

.0867484 

1.2184185 

.2837413 

.3/250/9 

.9853918 

•7505785 

73 

18 

.0970806 

I.I95I320 

.2987832 

.3883223 

.9833316 

.7568267 

72 

19 

.1079625 

1.1716702 

.3135459 

.4034387 

.9817725 

.7630174 

71 

20 

.1193876 

1.1480497 

.3280146 

.4178564 

.9798155 

.7691489 

70 

21 

.1313487 

1.1242896 

.3421750 

•4315745 

.9777607 

.7752204 

69 

22 

.1438386 

1.1004046 

.3560130 

.4445923 

.9756082 

.7812309 

68 

23 

.1568495 

1.0764112 

.3695144 

.4569094 

.9733584 

.7871798 

67 

24 

.1703731 

1.0523289 

.3826650 

.4685270 

.97IOI22 

.7930653 

66 

25 

.  I  844020 

1.0281752 

•3954514 

.4794460 

.9685698 

.7989851 

65 

26 

.1989265 

1.0039628 

.4078597 

.4896654 

.9660318 

.8046422 

64 

27 

.2139380 

o  9797109 

.4198760 

.4990726 

.9634002 

.8105204 

63 

28 

.2294272 

•  95544" 

.4314897 

.5080171 

.9606691 

•8159544 

62 

29 

.2453842 

.9311647 

.4426849 

.5161530 

.9578462 

.8215086 

61 

30 

.2617994 

.9068996 

.4534498 

.5235988 

.9549298 

.8269933 

60 

31 

.2786605 

.8826632 

.4637722 

.5303574 

.9519194 

.8324079 

59 

32 

.2959628 

•8584731 

.4736395 

.5364335 

.9488167 

.8377498 

58 

33 

.3136896 

.8343410 

.4830396 

.5418275 

.9456220 

.8432167 

57 

34 

.3318318 

.8102883 

.4919608 

•5465465 

.9424352 

.8482206 

56 

35 

.3503779 

.7863297 

.5003915 

.5505941 

.9389574 

•8533443 

55 

36 

.3693163 

.7624804 

.5083202 

.5539746 

.9354896 

.8583937 

54 

37 

.3886352 

.7387573 

.5157363 

•5566936 

.9319313 

.8633671 

53 

38 

.4083222 

•7I5I757 

.5226286 

•5587567 

.9282843 

.8682632 

52 

39 

.4283649 

.6917516 

.5289864 

.5601695 

.9245487 

.8730820 

51 

40 

.4487501 

.6685000 

•5347999 

.5609380 

.9207255 

.8778223 

50 

4i 

.4694661 

.6454363 

.5400590 

.5610692 

.9168148 

.8824831 

49 

42 

.4904986 

.6225756 

.5447538 

.5605695 

.9128181 

.8870639 

48 

43 

.5118340 

•5999330 

.5488748 

.5594464 

.9087354 

.8915636 

47 

44 

.5334592 

.5775230 

•5524133 

.5577075 

.9045682 

.8959812 

46 

45 

.5553604 

.5553604 

•5553604 

•5553604 

.9003163 

.9003163 

45 

sin  x 

x  sin  x 

jr  COS  X 

X 

JC 

TABLES. 
TABLE   XXIX   (WiNKLER). 


351 


- 

x  sin8  x 

X  COS"  4- 

x  sin  x  cos  x 

o 

o 

.5707963 

O 

O 

O 

O 

go 

I 

0.0000053 

•5528336 

0.0174480 

O.OOO473I 

0.0003046 

O.O27IO55 

89 

2 

.0000425 

•5340191 

.0348641 

.0018707 

.0012175 

.0535055 

88 

3 

.0001434 

•5142774 

.0522165 

.0041591 

.0027366 

.0793599 

87 

4 

.0003397 

•4936797 

.0694735 

.0073037 

.0048580 

.1044483 

86 

5 

.0006629 

.4722610 

.0866036 

.0112691 

.0075768 

.I288o6l 

85 

6 

.0011445 

•4500577 

•1035753 

.0160187 

.0108862 

.1524072 

84 

i 

.0018145 

.4271082 

.1203585 

.O2I5I52 

.0147782 

.1752269 

83 

8 

.0027036 

.4034495 

.1369227 

.0277206 

.0192432 

.1967883 

82 

9 

.0038440 

.3795486 

.1532356 

.0345961 

.0242696 

.2184312 

Si 

ro 

.0052628 

.354i6ii 

.1692701 

.0421025 

.0298469 

.2387751 

80 

ii 

.0069898 

.3286104 

.1849964 

.0501998 

•0359597 

.2582557 

79 

12 

.0090536 

.3025090 

.2003859 

.0588477 

.0426032 

.2768568 

78 

13 

.0114814 

.2758979 

•2I54"3 

.0669182 

.0447316 

.2898545 

77 

14 

.0143007 

.2488180 

.2300454 

.0776321 

•0573578 

.3"3654 

76 

15 

•0175372 

.2213107 

.2442622 

.0876861 

.0654498 

,3272492 

75 

16 

.0212165 

.1934174 

.2580362 

.0981263 

.0739907 

,3422069 

74 

17 

.0253628 

.1651794 

.2713432 

.1089108 

.0829579 

.3562311 

73 

18 

.0299996 

.1366392 

.2841597 

.1199979 

.0923291 

.3684670 

72 

19 

•035I491 

.1078370 

.2964635 

.1313468 

.1020806 

.3814589 

71 

20 

.0408330 

.0788151 

.3082329 

.1429154 

.1121876 

.3926566 

70 

21 

.0470712 

.0496147 

•3194479 

.1546625 

.1225963 

.4029094 

69 

22 

.0538829 

.0202775 

.3300896 

.16654/2 

.1333648 

.4122188 

68 

23 

.0612860 

0.9908418 

.3401397 

.1785287 

.1443808 

.4205874 

67 

24 

.0692976 

.9613504 

•3495819 

.1905671 

•1556439 

.42802O7 

66 

25 

•0779317 

•9318432 

.3584015 

.2026227 

.1671250 

.4345256 

65 

26 

.0872036 

.9023558 

.3665819 

.2146552 

.1787939 

.4401084 

64 

27 

•0971256 

.8727278 

.3741122 

.2265/43 

.1906187 

.4446770 

63 

28 

.1077095 

.8436043 

.3809827 

.2384996 

.2025721 

.4485524 

62 

29 

.1189646 

.8144150 

.3871810 

.2502359 

.2146179 

.4514376 

61 

30 

.1308998 

.7853982 

.3926990 

.2617994 

.2267254 

.4534498 

60 

3i 

.1435218 

.7565900 

.3975304 

.2731543 

.2388603 

.4546051 

59 

32 

.1568363 

.7280265 

.4016691 

.2842664 

.2509907 

.4549215 

58 

33 

.1708476 

.6997373 

.4051110 

.2951004 

.2630822 

.4544147 

57 

34 

.1855580 

•6717595 

.4078540 

.3056249 

.2751010 

•4531075 

56 

35 

.2009685 

.6441236 

.4098967 

.3165358 

.2870128 

.4510202 

55 

36 

.2170789 

.6168597 

.4H2396 

.3256181 

.2987832 

.4481748 

54 

37 

•2338865 

.5899977 

.4"8853 

.3350265 

.3103779 

.4445962 

53 

38 

.2513883 

.5635661 

.4118369 

.3440050 

.3217622 

.4403062 

52 

39 

.2695787 

•5375920 

.4110997 

.3525260 

.3329020 

•4353334 

5i 

40 

.2884511 

.5121006 

.4096806 

.3605649 

.3437628 

.4297035 

50 

41 

•3079975 

.4871170 

.4075877 

.3680945 

.3543106 

.4234443 

49 

42 

.3282075 

.4626638 

.4048309 

.3750942 

.3645114 

.4165892 

48 

43 

.3490699 

•4387632 

.4014216 

.3815415 

.3747317 

.4091532 

47 

44 

•3:05725 

.4154354 

•3973728 

.3874163 

.3837385 

.4011812 

46 

45 

.3926991 

.3926991 

.3926991 

.3926991 

.3926991 

.3926991 

45 

x  sin8  x 

_r  cos8  x 

.r  sin  x  cos  x 

* 

352 


A    TREATISE   ON  ARCHES. 


TABLE   XXX. 
STONE  ARCHES.* 


Name,  etc. 

Date. 

f 

Max. 
Span. 

Rise. 

4>- 

Reference. 

1905 
i9°3 
1380 
19031 
1859 
1893 
1901 
1902 
1888 
1833 
1833 
1889 
1901 

292. 
75- 
Si- 

r  3°. 
20. 
13- 

b  09. 

00. 

?S9-i 
101.8 
87-8 

57-3 
59-0 
S2.S 
21.4 
90.2 

4-9 
4-8 
4.0 
4.9 

tt 
6.6 

3-4 

4-5 

B    Z-28--04 

N.    IO-I2-OI 
3.    l2-7-'93 
N.   io-i7-'o3 
V.   7-29-'  99 
3.    1  2-7-'  93 

SI.    2-l8-?02 
SI.    1  0-4-'  02 
3.     2-27--02 
3.    5-2-'o2 

>.'    '92 

B.     I2-26-'0! 

A.  6-44 
P.   '52-294 
EL 

Blue  prints 
B.    i2-7-'93 
F.   '52-276 
F.  '52-290 

Luxemburg,  Germany  H. 

'  Italy                                         R 

Cabin  John,  Washington,  D.  C.  .  .  .  .  Aq. 

Black  Forest,  Germany  R. 
Bogenhausen,  Bavaria  H. 
Lavaur,  France  R. 
Grosvenor,  Chester,  England  H. 
Turin,  Italy  H. 

96. 
87. 
83- 
80. 
60. 
60. 
60. 

52.8 
SS-8 
f6o.o 
90.0 
t6o.4 
65  .0 
44.0 

S-6 
5-9 
S-3 
t6.o 
tS-9 

ll:l 

Vieille  Brioude,  France  
Ballochmoyle,  Scotland  R. 

1775 
1793 
1856 
1892 
1893 
1  545 
1830 

Gignac,  France  H. 
Victoria   Low  Lambton,  England.  ..R. 
Main  St.,  Wheeling,  W.  Va  H. 
Janma,  Austria  R. 
Tournon,  France  H. 
London,  England  H. 

59- 

57- 
56. 
52. 

'!f:i 

'6S:o' 

37.7 

1:1 

2.8 

Berne,  Switzerland  
Gloucester,  England  H. 

1204? 
1827 
1886 
1611 
1898? 

"i*34 
1336 

SO. 

B.    I2-I9-'9S 
F.   '52-290 

54-0 

4-5 

Near  Grenoble,  France  H. 
^ellefield    Pittsburg   Pa  H. 

S° 
SO 
48 
48 
47 

54-4 
36.6 

49-  5 

73^8 

3-2 

4.0 
5-6 
4-9 

4.6 

5-3 
4-5 

K.  '52-276 

B.     6-2  2-'  90 

I.  5    94 

B.   i2-7-'93 
F.   '52-276 
F.   '52-274 
B.   io-27~'o4 
K.  5-i7-'9S 

L  and  Q. 
P.   '96-140 

Engineer,  '04 
G.  '97-179 
G.  '88-575 
F.  '52-280 
F.  '52-276 
P.  '96-130 
F.   '52-296 
F.  '52-117 
B.  5-18-93 
B.   i2-7-'s>3 
L. 
B.   6-19-02 
B.   7-6-'  9  3 

F.   '52-276 
F.  'q  2-280 

Moulins   France                            ...    .    H. 

Verone  'italy  H. 

1354 
1904 

,T 

I8SS 
1755 

1899 

«.  46 
b  44- 

44- 

35- 

35-8 
19-3 

Outer  Maximilian,  Bavaria.  .  ,  H. 
Putney  Road,  England  H. 
Narni,  Italy  H. 

Alma,  Paris,  France  H. 
Pont-y-tu-prydd,  South  Wales  H. 

40 
40 

35-0 

2-5 
4-0 

C  33.  Bellows  Falls,  Vt  H. 
Albula  River  Viaduct  R. 
Verdun,  France  H. 

189? 
1884 
1766 
1732 
1203 
i8so1 
1849 
rS9o 
1893 
1  80  if 

3 

34-S 
34-5 
33-8 
32.6 
32.0 
31-2 
31. 
31- 
31- 

30.1 
t42.6 
38.2 
66.3 

43-7 
16.4 
16.4 
32.8 

3-9 
5-6 

7-7 
5-3 

5-9 

2.6 

it 

Waldi-tobel,  Bludenz,  Austria  R. 
Vizille,  Grenoble   France  H. 
Villeneuve,  France  H. 
St.  Martin,  Toledo,  Spain  H. 
Scrivia,  Italy  R. 
Viaduct  de  Moret,  France  R. 
Boucicault,  Verjue,  France  H. 
First  Worochta,  Austria  R. 
Aberdeen,  Scotland  H. 

Wan  Hsien,  China  H. 
North  Ave..  Baltimore.  Md.§  H. 
Echo  Bridge,  Newton  Upper  Falls, 
Mass  Aq.&H. 

1895 

1876 
1765 
1774 

30. 

29.0 
28.2 
28.2 

65.0 
26.0 

42-3 
38-5 

32.0 

5-0 

S.o 
6.4 
<;.  * 

Neuilly.  France  ...                   .  .  .  H. 

*  For  data  for  about  five  hundred  masonry  arches  see  "Symmetrical  Masonry  Arches.' 
by  Malverd  A.  Howe.  John  Wiley  &  Sons.  N.  Y. 

t  About.         J  27  B.C.-I4  A.D.          §  Brick  Ring.         a.  Two  hinges.         b.  Three  hinges- 


TABLES. 


353 


TABLE   XXX— (Continued). 
STONE  ARCHES. 


Name,  etc. 

Date. 

°6& 

Max. 
Span. 

Rise. 

to- 

Reference. 

Maidenhead,  England  R. 

1838 
1785 

6 

28.0 
27   6 

24.3 
63  8 

5-3 

K.  io-2S-'9S 

Near  Oisilly,  France  R. 

7 

27.0 

46.0 
6   9 

[  '   I2   7     93 

Tetes,  France  H. 
Devil's  Bridge,  Lucca,  Italy  H. 
Waterloo  (new),  London  H. 
Hartford,  Conn  H. 
Tongueland,  Scotland  H. 
Napoleon,  Paris,  France  R. 

1732 

riooo 
1817 
^1904 
1806 

5 
9 
8 

7 

23.6 
20.5 

20.0 
19.0 

18  .0 
16.0 

6r.8 
60.3 
34-6 
29.8 
38.0 
14.8 

4-7 
t4.5 
4-8 

"3.6' 

F.  '52-278 

F.  '52-288 
T.   2-i9-'o4 

L  and  F. 

Nantes,  France  '....'  H. 
St.  Esprit,  France  H. 
Second  Worochta,  Austria  R. 

1765 
1309 
1893 

3 
19 

iS-4 
14.1 
13.5 

34-0 
44-8 
56.8 

6.4 
5-9 

T     j  A  93 

i  and  Q. 
F.  '52-274 

Toulouse,  France  H. 
Lodi  St.,  Elyria,  Ohio  H. 

1632 
1894 

7 

13-0 

38.4 

3-7 

3ity  Engineer 

Sault,  France  
Winstone.  England  

1827 
1762 

3 

08.8 

Wurtemberg,  Germany  H. 
Balersbronn,  Germany  ?H. 

1882 
1889 

i 

08.8 
b  08.2 

10.8 
10.8 

3-3 

2.O 

G.  '91-903 
G.  '01-38 

Hartford,  Conn  H. 
Orleans   France   .                        H. 

11904 
1760 

8 

L  and  F 

Ponthaut,  France  

1793 

06.3 

53-i 

5-7 

F.  '52-284 

Wissahickon,  Philadelphia,  Pa  H. 
Potomac  Aq.,  Georgetown,  D.  C.  ..Aq. 

1897 

7 

05.0 

B.  9-9-'  9  7 
A.  '37-148 

Prague,  Bohemia  H. 
Herault,  France  
Port-de-Piles   France                             R 

1878 
1847 

7 

i 

04.4 
03  8 

ti6.o 
15-4 

4.0 
2.7 

K.  s-io-'78 
F.  '52-280 

Avignon,  France  ;  H. 
Gere,  Vienna,  Austria  
Munich,  Bavaria  H. 
Pont-de-la-Concorde,  Paris  H. 
Guillotiere,  Lyons,  France  H. 
Pont-au-Double  ,  Paris,  France  H. 

1187 
1781 
1814 
1792 
1265 
1847 

21 

3 

02.9 
02.7 
02.3 
02.3 
02.3 
or.8 

Si-5 
28.2 

'5:5 
11 

2.4 

5-2 

4-3 
3-7 

2.  I 

5-3 

J,  L,  and  F. 
F.  '52-284 
F.  '52-288 
J,    F.  '5  2-284 
F.   '52-274 
J,  F.  '52-296 

Goltzsch  Viad.  Bavaria  j  R. 
Wellington,  Leeds,  England  H. 

1851 
1819 

6 

01.7 

00.0 

50.9 

7-4 

Q.  Am.  Supp. 
A.  '44-128 

Blackfriars  (old),  London,  England  .H. 

1770 

9 

00.0 

43-0 

5-0 

L,  F.'  5  2-280 

Bishop  Aukland,  England  H. 

1388 

1.8 

I. 

Alcantara  Aq.,  Lisbon,  Spain  Aq. 
Rutherglen,  Scotland  H. 
Minneapolis,  Minn.  .  .                        .  .  .R. 

ti775 
1895 
1883 

35 
3 

00.0 
00.0 

88.0 

12-5 
39.7 

4.0 
3.0 

K   K^ls 

t  About. 


J  Brick  ring. 


b  Three  hinges. 


354 


A   TREATISE    ON  ARCHES. 

TABLE  XXX— (Continued). 
PLAIN  CONCRETE  ARCHES. 


Name,  etc. 

Date. 

0) 

Max. 
Span. 

Rise. 

to- 

Reference. 

Ulm,  Wiirtemberg,  Germany  H. 
Neckarhausen,  Germany  H. 

{1903 

6187. 
6165. 
bi6s 

8.7 

1:1 

3.6 

K    ?;-30-°04 
B    o—  26—  '01 

Munderkingen,  Germany  H. 

'1893' 

bi64. 

6.4 

3.3 

Q             Y 

Connecticut    Ave.    Bridge,    Washing- 
ton  D  C                                              H 

5  •  o 

N    »_g_'os 

Vauxhall,  London,  England  H. 
Inzigkofen   Germany. 

ti849 

l8o6 

6144. 
6141 

1  1:? 

iV      '                   . 

N.   2-25-  99 

Big  Muddy,  111.  Cen.  Ry  R. 

140. 

0    0 

5-° 

Coulouvreniere,  Geneva,  Switz  H. 

1895 

8^2 

3  .  0 

Y'.  *9»I2~°3 

Borrodate   Burn  Viad.,  Scotland  R. 

1899 

12     . 

2.  5 

4-0 

B.    2-9-'  99 

Sixteenth  St.,  Washington,  D.  C..  .  .H. 

1905 

12     . 

9-0 

5-0 

B.   lA^'os 

Kircheim    WUrtemberg                          H 

•(•1898 

fc}l    4    6 

t   9  .0 

2.6 

Grand-Maitre,  France  Aq. 
Worms   Germany                        H. 

1869 

1900 

fen 

t   9-3 

K.  io-?6900 

Mittenberg,  Germany  H. 

Danville    111                                               R 

1899 

bn    . 

6!4' 

2.5 

B.    7-25-'oi 

Near  Mechanics  ville,  N.  Y  E.R. 

I9°5 

100. 

N.  3—  3—  '06 
B.    u-5-'o3 

Thebes,  bridge  approach  R. 
Schlitza  River,  Austria  H. 

ti903 

1 

100. 

0.0 

o  .  o 

4.5 

2  .  3 

B.   s-n-'os 
K.  4—  12—  '04 

Silver  Lake,  Pittsburg,  Pa  R. 
Imnau,  German  v  H. 
Morar  Viaduct,  Scotland  R. 
Santa  Ana  Viaduct,  Riverside,  Cal.  .R. 

1905 
1896 

1904 

100. 

b  98. 
90. 
86. 

0.0 

9.8 
24.0 
43-0 

4.0 
1.5 

3-5 

N.  5-6-'  05 
G.   'ox-40 
B     2-9-99 

N.  9-9-05 

San  Leandro,  Cal  H. 

1901 

Piano,  111  R. 
Rechtenstein,  Wiirtemberg  

1904 
1893 

,    75- 
b  74. 

'  'k'.'a 

3-0 

N.  6-  a-'  04 
Y.  '98 

Ashtabula,  Ohio  R. 

1904 

74. 

37.0 

1*6   s 

T.    2-27-05 

Ehingen,  Wiirtemberg  H. 
Bridge  No.  163,  W.  Cincinnati,  O..  .  .R. 

1898 
1904 

b   69. 
68. 

17.0 

3-5 

B.     2-9-'  02 

N.  3-5-04 

Concord,  Mass  H. 

1901 

66. 

I  I  .  O 

Thebes  bridge  approach  (east)  R. 
Bridge  No.  242,  W.  of  Cincinnati,  O.  R. 

1905 
1904 

V 

65- 
60. 

32-5 
26.0 

3-3 

2.7 

B.     II-20-'02 

N.  3-5-04 

t  About. 


*  Clear. 


b  Three  hinges. 


SUPPLEMENTARY  TABLE. 
PLAIN  CONCRETE  ARCHES. 


Name,  etc. 

Date. 

~£ 

J! 

Max. 
Span. 

Rise. 

to. 

Refer  e»ce. 

Nashua  to  Hudson  N  H  H 

6 

Scotland,  Pa  R. 

B.     4-  S-'og 

Near  Guggersbach,  Switz  H. 
Wiesen  Switz  .  .  R 

1906 

169 
1  80    4 

Is 

3-7 

B.     4-30-'o8 
N    n-2i-'o8 

Lautrach,  Bavaria  R. 

^187.5 

B.     5-  2-'o7 

Mannheim,  Germany  H. 
Near  Kempton,  So.  Germany  R. 

Walnut  Lane,  Phila.,  Pa  H. 
Between  Teufen  and  Stein,  Switz  H. 
Rocky  River,  Cleveland,  Ohio.  .E.  &  H. 
Munroe  St.,  Spokane,  Wash..  .  .E.  &  H. 

1908 
1906 

1908 
1908 
1909 
19091 

4 
6 

0 

4 

6192 

6211.5 
233 

22L9'3 

281 

18.1 
86.7 
70.3 
'to.9 

us 

4.4 
9-5 

6!o 
6.8 

B.     6-i8-'o8 
j  N.    2-23-'o7 
IN.    s-n-'o? 
j  B.    I-3I-'07 
IN.    2-i5-'o8 
B.  12-  3-'o8 
N.     i-23-'o9 
B.     9-  2-'o9 

t  About. 


b  Three  hinges. 


TABLES. 


355 


TABLE  XXX— (Concluded). 
REINFORCED  CONCRETE  ARCHES. 


Name,  etc. 

Date. 

I* 

Max. 

Span. 

Rise. 

to- 

Reference. 

Munich,  Germany.  H. 

1904 

6230.0 

B.  ii    17  '04 

Decize,  France  H. 
Bormida  River,  Italy  H. 

1902 
1899 

183.7 
167.3 

5-3 
6.7 

1.6 

2  .O 

G.  '05-292 
G.  [04 

Playa-del-Re'y  Cal  H. 

|  ' 

Vj.     oc   377 

Schwinmschulbrucke,  Steyr  H. 
Park  Ave.,  Newark,  N.  J  H. 
Kansas  Ave.,  Topeka,  Kan  H. 
Zanesville,  Ohio,  V.  Br  H. 
Wildegg  Route,  Switzerland  H. 
Washington  Ave.,  Lansing,  Mich.  .  .H. 
Jacaguas  River,  Porto  Rico  H. 
Yellowstone  National  Park  H. 
Milwaukee  Wis  H 

1905 
1898 
1902 
1890 

1901 
1903 

138.4 
132.0 
125.0 

122.0 
122.0 
120.0 
120.0 

Jiaoio 

118  o 

9.4 
16.2 

18.9 

n-5 
11.4 

*|i:? 

2.0 
2-5 

0.6 

2.3 

2.0 

N.  8-i2-'oS 
B.  4-2-;96 

Cement,  3-'o2 
N.  8-3-'oi 
B.    i-I4-'p4 

Third  St.,  Dayton,  Ohio  H. 
Green  Island,  Niagara  Falls,  N.  Y.  .  H. 
Laibach  Austria  H 

1901 

IIO.O 

14.3 
ii  . 

2.  I 

3-3 

ft   3-4-'o~4°S 
B.    i2-6-'oo 

Route  Francois-  Joseph,  Austria.  .  .  .  H. 
Stockbridge,  Mass  H. 
N.  6th  Ave.,  Des  Moines,  Iowa  H. 
Wayne  St.,  Peru,  Ind  H. 
Bridge  113,  near  Marshall,  111  R. 
Yorktown,  Ind  H. 
W  St  Paterson  N  J  H 

1900 
1895 
TI902 
1905 

tioos 
1898 

6108.  2 

100.  0 
100.  0 
100.  0 
95-0 

14.. 

20. 
IS- 
40. 

1.6 
o  .  8 

4.0 

j.   '04 
B.    i2-7-'95 
Cement,  7-'  02 
B     3-29-'o6 
Slue  prints 

Main  St.,  Dayton,  Ohio  H. 
Grand  Rapids,  Mich  H. 
Seeley  St.,  Brooklyn,  N.  Y  H. 
Route  Payerbach,  Austria  H. 
Fabriano  Viaduct,  Italy  H. 
Route  de  1'Empereur,  Bosnie  H. 
Haldu  H 

1903 
1904 
1904 
1900 
1905 
1897 

88.0 
87.0 
85-3 
85-3 
84.9 
83.2 

t8. 

"s'. 

5- 
26. 
8. 

i-3-5 
i-S 

2.0 
I  .0 

3.   5-19-04 
3.    i2-i-'o4 

5-  "-i"31' 

N.    12-9-05 

Soissons,  France  H.  &  R. 
Rock  Creek,  Washington,  D.  C  H. 
La  Salle  St.,  So.  Bend,  Ind  H. 
Wishawaka,  Ind  .H. 
Hyde  Park  on  Hudson  NY...  H 

[•1903 
1901 

1*1906 
1807 

81.0 
80.0 
79.0 
76.0 

8.1 
14.0 
12.5 
15.8 

1.7 
i.S 

i.   10-7-  04 
1.    10-31-01 
Blues 
Blues 

Hamilton  St.,  Hartford,  Conn  H. 
Grand  Rapids,  Mich  H. 
Polasky  Cal  .  H 

1898 
1905 

75-0 
75-0 

7-5 
14.0 

1-3 

3.    3-22-'o6 
N      2-24-'o6 

Wabash'  Ind  H. 
Illinois  St.,  Indianapolis,  Ind  H. 
Meridian  St.,  Indianapolis,  Ind  H. 
La  Salle,  111  H. 

1905 
1900 
1900 
1905 

0000 

18.0 
9-5 
9-5 

7-5 

1-3 
1-3 

'2.0 

^J.  12-2-05 
3.  4-n-'oi 

3.  9-2i-'os 

Route  de  Pa'inperdu,  Belgium  H. 
Near  Copenhagen,  Denmark  H. 
Trinidad,  Col  H. 
Eden  Park,  Cincinnati,  Ohio  H. 
Bloomfield  Ave.,  Newark,  N.  J  H. 
Guayo  River,  Porto  Rico  H. 

1899 
1879 
1905 
1895 
1904 

71.8 
71  •  7 
70.0 
70  .0 

7.0 

10.  0 

8.5 

I  .2 

G.  '04 
3.  7-21-98 

"f.  2-io-'o6 
J.    io-3-'95 
sT.  8-12-05 
N.  8-3-'  oi 

Auch,  France  H. 
Jacksonville,  Florida  H. 
Herkimer  Viaduct,  N.  Y  R. 

1899 
1904 

I 

68.9 
66.0 
66.0 

6.8 
t?.o 

1.0 

>.  '04 

Route  Cantal,  Italy  H. 

1902 

65.6 
65   6 

6.6 
14  8 

i   6* 

>.  '04 

G.  "04 

Route  Ebhausen,  Wlirtemberg  H. 
Troy,  N.  Y  H. 
Montgomery  St..  Jersey  City,  N.  J.  .H. 
Franklin  Bridge,  St.  Louis  H. 
Eighth  Ave.,  Carbondale,  Pa  H. 

1891 
1897 
1896 
1898 
1896 

65.6 
65.0 
61.2 
60.0 
58.7 

8.2 

8.5 

12.0 

'5-5 

6.6 

I  .0 

I  .0 

0.9 

G.  -04 
J.  i2-io-'98 

t  About. 


t  Center  to  centre. 


b  Three  hinges. 


356 


A  TREATISE   ON  ARCHES. 


SUPPLEMENTARY   TABLE. 
REINFORCED   CONCRETE  ARCHES. 


i/i 

Name,  etc. 

Date. 

y 

Max. 

Span. 

Rise. 

to 

Reference. 

Broad  St.,  Bethlehem,  Pa  E.  &  H. 

1910 

5 

100 

J  Cement  Era, 
I         8-'o9 

Near  Nashville  Term              H. 

1910' 

i 

IOO 

N.  ii-  s-'io 

Mulberry  St.,  Harrisburg,  Pa.  \p   a.  TJ 
Cameron  St.  approach       J      K 

1909 

29? 

IOO 

14 

1.5 

f  N.  8-iS'-o8 
\N.     4-3-09' 

Ross  Drive,  Washington,  D.  C  H. 
Vermilion  R.,  Danville,  111  R. 

1908 
1905 

i 
3 

6100 

00 

15 
40 

2-5 
4  -0 

3.     s-2i-'o8 
T.      7-i3-'o6 

No   113   Marshall   111                            R 

oo 

4  •  ° 

Blue  prints 

Highland  Boulv.,  Milwaukee,  Wis.  .  .H. 

1909 

2 

oo 

21.7 

i  •  7 

N.     6-12-09 

Cumberland  R.,  K.  &  T.  Ry  R. 

1907 

5 

02  .  I 

T.       3-22-*07 

Red  Bridge,  Huntington,  Ind  H. 

1907 

2 

05 

14 

1.8 

J  Cement  Age, 
\         io-'o8 

Pelham,  Borough  of  Bronx,  N.  Y  H. 

1908 

6 

05 

16.5 

2  .0 

N.  io-3i-'o8 

Factory  St.,Canal  Dover,  Ohio.E.  &  H. 
San  Diego  Cal                        ....    '      H 

1906 
1910 

3 
6 

06.7 
b  07 

ii.  8 

2  .0 

N      2-  9-'o7 
N.    3—  25—  ?u 

Paterson,  N.  J            H. 

1907 

3 

08 

12 

2  .3 

N.     3-  7-'o8 

Cedar  St    Mishawaka,  Ind  H. 

1  903 

3 

10 

14 

N.     7-  7-'o6 

Indianapolis,  Ind  H. 
Jefferson  St    South  Bend  Ind       .  .   H 

1905 

5 

10 
IO 

14-3 

14   . 

2  .  I 

2  .  3 

Blue  prints 

N.     7—  28—  '06 

igoSt 

B         4—  22  —  'oo 

Austin,  Texas  H. 

1910 

8 

15 

ll" 

B!     6-23-'io 

Paulius  Kill,  Near  Portland,  Pa  R. 

1909! 

7 

20 

60 

6  .0 

N.     7—  16—  *io 

Bridge  St.,  Peoria,  111  H. 

1907 

4 

25 

2-5t 

/  Eng.  World, 
1         2-'o7 

St.  Paul   Minn                      H 

1909 

25 

_ 

ft      4_  ,_-00 

Charles  River,  Boston,  Mass.  .  .H.  &  E. 
Bannock  St.,  Denver,  Colo  H. 

1910 
1907 

10 

i£3 

19. 
13. 

4-5 

2  .0 

XT'                   ^     ,     " 

N.  12-17-  10 
N.     3-2i-'o8 

Baltimore,  Md  H.  &  E. 

1909 

4 

39 

43  . 

3-8 

N.     6-i9-'o9 

Grand  Ave.,  Milwaukee,  Wis  H. 

1910 

10 

45 

28. 

2  .  2 

N.     2-  9-'o7 

Playa  del  Rey.  Cal  H. 
St.  Jean  La  Riviere,  Viaduct  E. 

1906? 
1908 

46 
a  49.2 

18. 

40. 

2  .0 
I  .9 

N.    3-31-06 
N.  12-  3~'io 

Delaware  R.  near  Portland,  Pa  R. 

1910 

9 

50 

B.  12—  30—  '09 

Grand  R..  Painsville,  Ohio  R. 

1908 

3 

60 

s&'. 

7  -3 

N.     5-  i-'o9 

Tavanasa,  Switzerland  H. 

1908 

i 

b  67.3 

18. 

0.7 

B.     2-i8-'o9 

Pyrimont,  France  H. 
Guindy  River  Bridge,  France  R. 
Eel  River  Bridge,  Humboldt  Co.,  Cal.H. 

1907 
1907? 
igut 

3-5 

i 
7 

,    69 
b   77    . 
So 

25- 

21. 

26. 

s'o 

B.     4-  2-'o8 
B.     3-26-'o8 
B.     3-23-'i  i 

Berlin,  Germany  H.  &  R. 

1910 

3 

83.7 



N.     8-13-10 

Decize,  France  H. 



83.7 

IS. 

"i'.6' 

G.  4  Tri.  1905 

Meadow  St.,  Pittsburgh,  Pa  H. 

1910 

7 

09 

46. 

6.2 

B.    12-    I-'lO 

Sitter  Bridge,  Switzerland  H. 

1908 

7 

59 

87. 

3-9 

V.     3-13-09 

43d  St.,  Philadelphia,  Pa  H. 
Grafton  Bridge,  Auckland,  N.  Z  H. 

1909 
1910 

i 
i 

62 

b320 

'B: 

B.     s-2o-'o9 
B.     8-  4-'io 

Hudson  Memorial  Br.  (Proposed)  

i 

703 

177. 

15-0 

N.  n-i6-'o7 

t  About. 


t  Clear. 


a  Two  hinges 


b  Three  hinges. 


TABLES. 


357 


TABLE  XXXI. 
CAST-IRON  ARCHES. 


Name,  etc. 

Date. 

Engineer. 

"Z, 

Span. 

Rise. 

Reference. 

Southwark,  London,  England.  . 
Sunderland,  Durham,  England. 
St.  Louis,  Paris,  France  
Rock  Creek,  Washington,  D.  C. 
El-Kantara,  Algeria  
Pont  du  Carrousel,  Paris  

1819 
1796 
1862 
1858 
1864 
1836 

Rennie 
Burdon 
Martin 
Meigs 
Martin 
Polenceau 

3 
3 

*240 

236 

210 
200 

34 

26 
6 

A.  '56-261 
A.  '42-334 
Iconographic 
K.  5-3-67 
K.  3-8-'  67 
A.  '39-81 

Blackfriars,  London  

1869 

Cubitt 

5 

*i86 

Chestnut  St.,  Philadelphia  
Staines,  England  

1866 
1803 

Kneass 
Wilson 

185 
1  80 

•' 

K.  6-26-'  68 
K.  9—  13—  '95 

Galton 

1  80 

Lendal    York   England 

1862 
1826 

Page 
Telford 

172 
170 

\.  Vol.  25185 

Tewksbury,  Gloucester,  Engl'd. 

Battersea,  England  
New  North,  Halifax,  England.  .  . 
Hill's,  Bristol,  England  

1890 
1869 
1809 

Bazalgette 
Fraser 

5 

2 

*i63 
1  60 
1  60 

K.  4-i9,-'9S 
K.  5-7-69 
A.     55-111 

France  

1899 

3 

"158 

^-   9-5~'o3 

Bonar,  Scotland  

1812 

Telford 

Craigellachie,  Scotland  

Telford 

1 

150 

5 

High  Bridge,  England  

l83O 

Potter 

j 

140. 

£"    '  37—  151 

Buildwas   England 

Telford 

Barnes,  England  
Victoria,  Windsor,  England.  .  .. 

1849 
1851 

Locke 
Page 

3 

*I2o! 

<••   5-3l-]9S 

Westminster  (new)  England  

1863 

Page 

'120. 

C.  3-8-'  95 

Chepstow,  England  

5 

*II2. 

Pont  d'Austerlitz,  Paris  

"1806' 

Lamand6 

5 

*I06. 

>. 

New  Leeds   England  

Steel 

i 

IO2. 

Coalbrookdale,  England  
Lary,  Plymouth,  England  

1779 
1827 

Darby 
Rendel 

I 
5 

l~ 

Bristol,  England  

1806 

Jessop 

*IOO. 

Nottingham,  England  
Richmond    England 

1871 
1848 

Tarbottom 
Locke 

3 

100. 

['  I°~61~'71 

St.  Denis    Paris  

:IOO  . 

1  1  . 

*     *             95 

Ravenswharfe,  Dewsbury,  Eng. 

1848 

Grainger 

2 

100. 

12. 

L.   '48-62 

W.  of  Leicester,  England  
New  Logan   Glasgow,  Scotland 

1839 
1890 

Vignoles. 

3 
4 

*  91  • 

is'. 

'•    3-25-'o4 

C.    8-22-',JO 

Witham,  Boston,  England  

Rennie 

86. 

5- 

0- 

Maximum. 


358 


A   TREATISE    ON   ARCHES. 


TABLE   XXXII. 
WROTJGHT-IRON  AND  STEEL  ARCHES. 


Name,  etc. 

Date. 

Engineer. 

"8  I 
dw 

£ 

Span. 

Rise. 

Reference. 

Clifton,  Niagara  H. 
Viaur  Viaduct,  France  R. 
Bonn,  Germany  H. 
Dlisseldorf  ,  Germany  H. 
Douro,  Portugal  H.  &  R. 
Kaiser  Wilhelm,  Germany.  .R. 
Niagara  H.  &  R. 

899' 
898 
898 
885 
897 
897 
884 

Krohn 
Krohn 
Seyrig 
Rieppel 
Buck 
Eiffel 

i 
3 
4 
6 

j 
7 

i 

a  840.0 
6*721.6 
a*6i4.o 
0*594-5 
566.0 
*   557.  6 
550.0 

76.2 
9S!4 

59.5 
31.5 
24.0 

R.  n-i-'Sp 
B.   8-8-'  95 
B.   4-2o-'99 
K.  7-2-'86 
N.  i2-25-'97 
B.   8-6-;  96 

Bellows'Falls,  Vermont  H. 
Levensau,  Germany.  .  .H.  &  R. 

90S 
894 
877 

Worcester 
Lauter 
Eiffel  &  Co 

i 
i 

b  540.0 
a  536.0 

T90  .0 
69.0 

N.  4-29-05 
K.  8-i6-'9S 

Eads,  St.  Louis  H.  &  R. 
Grlinenthal,  Germany.  H.  &  R. 
Washington  Br.,  New  York.  .  H. 
Victoria  Falls,  Africa  R. 

873 
892 
889 

Eads 
Eggert 
Hutton 

3 

i 

2 

*520.0 

a  513-5 

a  510.0 

T53-0 
77-3 
98.3 

Br.  History 
K.  8-i6-'95 
Br.  History 

Paderno,  Italy  R. 
Lake  St.   Minneapolis  H. 
Costa  Rica  R. 

888 
902 

Rothlisberger 
Sewall 
Cooper 

4 

492.0 
6*456.0 

123-0 

jgo.o 

B.  6-iI-;03 
N.   12-7-95 

Driving  Park,  Rochester,  N.  Y.H. 
Doran  Arch,  Richmond,  Ind.  H. 
Trisana,  Austria  R. 
Worms,  Germany  R. 

889 

882' 
900 

Buck 
Doran 
Huss 

3 

b  428.0 
b  408.0 
a   393-  6 
0*383.1 

t48   2 

B.    6-22-'99 

G.  '88xvi-73S 

Schwarzwasser,  Switzerland  R. 
Panther  Hollow,  Pitts.,  Pa..H. 
Kornhaus,  Berne,  Switz.  .  .  .H. 
Pont     Alexandre     III,     Paris, 

882 
'    898' 
896 

Probst 
Schultz 

Resal 

6 

373-9 
360.0 
*   376.7 

45-0 
103.7 

G.  '91 
N.  6-4-'98 
B.  i2-i6-'<>7 

Austerlitz,  Paris,  France.  .  ..R. 
Worms,  Germany  H.  ? 

90St 

i 

b  351.6 
0*346   4 

T40.0 

B.   12-7-05 

Troitsky,  St.  Petersburg,  Rus- 
sia    H. 

En    neer  'o 

Stony  Crk..  British  Columbia  R. 
Palatinus,  Mayence,  Germany  .  . 
Mirabeau.  Paris,  France.  .  .  .H. 
Pesth,  Austria  H. 
Coblenz,  Prussia  R. 
Foot-bridge,  Paris,  France.  .H. 
Germany  
Verona,  Italy  H. 

893 

885 
896 
873 
866 
t  882 

885' 
t  898 

Peterson 
Lauter 
Resal 
Gouin  Co. 
Hartwich 
Moisant 

Biadego 

i 
4 
3 
5 
3 

a  336.0 
0*335.0 
0*326.0 
0*321  .0 
315.0 
302.3 
a  298.5 
a  288.6 

80.4 
19.5 
28i6 

40.3 

J3S.V 

N.    2-20-!04 

K.  6-5-96 

K.  6-7-67 
K.  3-9-83 

K.  4-i7-'8s 

Viad.  de  Passy  Paris,  FranceR. 

a  281   i 

Arcole,  Paris,  France  H. 
Main  St.,  Minneapolis,  Minn  .  H. 
Paris,  France  H. 
White  Pass.  Alaska  R. 
Blaauw-Krautz,  Cape  Colony.  . 
D'Argenteuil,  France.  Aq.&  H. 
Versham,  Switzerland  H. 
El  Cinca,  Spain  H. 

855 
888 
899? 
900 
884 
894 
897 
866 

Oudray 
Strobel 
Lion 
Wood 
Max  Am  Ende 
Bechmann 
Berg 

2 

3 

3 

262.4 
b   258.0 
0*246.0 
b   240.0 
229.7 

*229.6 

229.6 

26.0 
49.3 
90.0 

F97-0 

22.  2 

B.   4-i4-'88 
N.   s-io-'90 
B.   8-30-00 
B.   3-28-01 
B.   1-24-85 
G.  '97 
Engineer    97 

Lafayette.  Lyons,  France.  .  .H. 
Moraude,  Lyons,  France  .  .  .  H. 
So.    Market    St.,    Youngstown, 
Ohio  H. 
Fraser  River.  Can.  Pac  R. 
Near  Iron  Mountain.  Mich.  .  .  R. 
Pont  du  Midi,  Lyons,  France.  H. 
Fairmount.  Philadelphia,  Pa.H. 
Croton  Dam,  N.  Y  H. 

889 
890 

898 
893 
902 

'    897' 

1"  9°S 

Tavernier 
Tavernier 

Fowler 
Peterson 
Loweth 

Thayer 
Smith 

3 
3 

I 
I 
I 
3 
4 

*22I  .  I 
*22I  .  I 

a  210.6 

210.0 
7     207.0 

1     200.0 

b    200.0 

I4.6 
14-6 

60.0 

l"47-  o 
14.7 
40.0 
43-4 

£•  '93. 

R.  9-91 

N.    2-4-99 
K.  n-29-195 

B.     11-20-02 

^enie  Civil  '92 
Blue  prints 

B.     I  2-1-'  04 

Noce  Chasm,  Austria  H. 
Canton  Berne,  Switzerland.  .  H. 
Forbes  St..  Pittsburg,  Pa  H. 
Cambridge  Boston,  Mass.  .  .H. 

890 
897 

90ot 

Hagen 
Jackson 

i 
i 
i 
ir 

a   196.8 
a  196.8 
)   195.0 

01*188.5 

33-9 
J32.8 
59.0 

26.7 

3.     2-I-'90 
3.     2-26-'03 

N.  4-i5-'°5 

Maximum. 


t  About. 


t  Clear. 


a  Two  hinges. 


b  Three  hinges. 


TABLES. 
TABLE  XXXII— (Continued). 

WROUGHT-IRON  AND  STEEL  ARCHES. 


359 


Name,  etc. 

Date. 

Engineer. 

c 

r 

Span. 

Rise. 

Reference. 

Blackfriars,  London,  EnglandR. 

1886 

Barry 

5 

*i8S-o 

18.5 

K.  2-i-'9s 

Fille,  France  H. 

1896 

i 

b  184.8 

16.4 

Anel  River,  Sumatra  R. 

Post 

a  184.0 

R.    1  1-'  97 

Trevallyn,     Launceston,     Tas- 

Doyne 

184  o 

Vienna,  Austria  R. 

1897 

Gridl 

j 

183.7 

Pimlico,  England  R. 

1860 

Fowler 

4 

a  175-0 

17-5 

K.  3-22-'9S 

New  North,  Edinburgh,  Scot.  H. 
Pont  Boiddien   Rouen,  Fr.  ..H. 

1899 
1888 

Blyth 

3 

ti75.o 

K.  1  0-6-'  99 

Becton,  England  '  Gas 

1870 

Evans 

i 

fil'.o 

£3'.° 

Vienna,  Austria  H. 

ti897 

Pfeuffer 

* 

b   174.0 

5 

N.  8-  1  8-'  oo 

Garibaldi,  Rome  H. 

1888 

Vescovali 

2 

ti6^3 

R.    I-'93 

Cerveyrette,  France  H. 
Fall  Creek,  Ithaca,  N.  Y.  .  .  .H. 

1892 
1898 

Baldy 
Landon 

I 

a  170.0 

137-8 
34-0 

R.    2-'92 

B.  4-28-98 

Chagrin     River,     Bentleyville, 

Ohio  H. 

1896 

Osborn 

1 

a   168.8 

29.5 

Osborn  Co. 

Manhattan  Arch,  N.  Y  H. 

tlQ02 

,Qf\A 



I 

a   168.5 

B.     2-IO-'03 

Brooklyn,  Ohio..  .  -  H. 
King  Charles,  Stuttgart,  Ger.  H. 
Mill  St    Watertown,  N.  Y..  .  H. 

I  OQ4 

1893 

Leibbrand 

5 

i 

0*165  .6 
b   165.0 

t«:? 

N.  3^-5-'98 

Canningtown,  England  H. 
Forbes  St.,  Pittsburg,  Pa  H. 

1897 
1874 

Binnie 
Pfeifer 

i 

Jtso.o 
£150.0 

1*5.1 

£26.0 

N.   7-iS-'99 

Cedar  Ave.,  Baltimore,  Md.  .  H. 

1891 

Latrobe 

b  150.0 

38.0 

T.    9-i8-'9i 

Battersea,  England  R. 

1863 

Baker 

5 

144.0 

K 

Forbes  St.,  Pittsburg,  Pa.  ...  H. 
Riverside  Cemetery,  Cleveland, 

1899 

Brunner 

a  144.0 

24.0 

N.   7-i5-'99 

Ohio  H. 

1896 

Osborn 

I 

a  142.0 

27.0 

Dsborn  Co. 

Anacostia,  Washington,  D.  C.H. 
Manhattan  Viaduct,  N.  Y.  .  .H. 

1900 

Douglas 
Williamson 

6 
23 

6tl29.  2 

*i   8.6 

ti4.5 

N.  8-i9-'os 
B.   6-8-'99 

Vienna  ,  Austria  R. 

1897 

Gridl 

3 

*i   9.4 

Albert,  Glasgow.  Scotland.  .  H. 
Victoria,  Stockton,  England.  H. 

1870 
1887 

Bell 
Neate 

3 
3 

*i   4.0 

K.  7-1-70 

Michigan  Ave.,  Lansing,  Mich.H. 
Parahyba  River,  Brazil  R. 

1895 

Landor 
Ellison 
Oldfield 

tio.5 

K.  8-2  1-'  68 

Mvtao...  '.'.'.'.'.'.!".'.'.".'.".".  ".".ft 

Mill  Creek,  Youngstown,  O  .  .  H. 

1894 

Page 
Fowler 

I 

Jioo.o 
b     96.0 

tio.5 

K.  8-7-68 
Fowler 

Carlsburg  Viaduct,  Denmark  H. 

a*  90.0 

J38.'o' 

ST.  5-i6-'o3 

Lake  Park,  Milwaukee,  Wis.  H. 

'1897' 

Sanne 

I 

a     87.0 

14.0 

Sanne 

Rock  Lane,   New   Haven,  Con- 
necticut    H. 

1891 

Hill 

I 

b     84.3 

14-3 

B.  8-1  6-V 

Weston   Aq.,    Southboro,    Mas- 

sachusetts    Aq.  &  H. 
Oker.  Brunswick,  Germany.  .  H. 

1903 

Stearns 
Haeseler 

I 

t  80.0 
b     78.7 

Uo 

V.    IO-25-'O2 
R..    2-'  89 

Richmond  Wier,  England.  .  .  H. 

i892J" 

More 

5 

t*  66.0 

EC.  6-28-'9S 

Thirtieth  St.,  Philadelphia,  Pa.  R. 
Lake  Park.  Milwaukee,  Wis.  H. 

1894 

Wilson 
Sanne 

i 
i 

i     64.1 
'     50.0 

'0:° 

£.  7-2  2-'  70 
Sanne 

t  About.  t  Clear.  a  Two  hinges. 

SUPPLEMENTARY  TABLE. 
WROUGHT-IRON  AND  STEEL  ARCHES. 


b  Three  hinges. 


Name,  etc. 

Date. 

Engineer. 

o'w 

Max. 
Span. 

Rise. 

Reference. 

2 

Rio  Fiscal  Br.,  Guatemala.  ..  R. 
Assopus  Viaduct,  Greece.  .  .  .R. 
Fort  Snelling,  Minn  H. 
Mannheim,  Germany  H. 

1907 
1906 
1909 
1908 

Vogue 
Contractors 
Shunk 

6364^0 

38.5 
78.0 
84.5 

22.8 

N.  4-4-08 

B.  1  1  -4-09 

N.  6-26-'o9 
B.  6-18-08 

Oakland  Br.,  Pittsburgh,  Pa.H. 

1907 

Whited 

440 

70.0 

B.  5-16-07 

Tonkin,  China  R. 

b  532.8 

94.8 

B.  5-27-09 

a  Two  hinges. 


b  Three  hinf 


A    TREATISE    ON   ARCHES. 


TABLE  XXXIII. 

METAL  ROOF-ARCHES. 


Name,  etc. 

Date. 

Span. 

Rise. 

Reference. 

Liberal  Arts  Bldg.,  Col.  Ex.,  Chicago...  .   b 

1892 

368.0 

206.3 

B.  9-i-'92 

Rpof  of  Main  Bldg.,  Lyons  Ex  a 

1894 

361.0 

108.0 

Train-shed,  Philadelphia,  Pa.,  Penn.  R.R.b 

1893 

300.7 

100.3 

B.  6-i-'93 

"Pa.&R.R.R6 

1892 

259.0 

JS8.3 

B.   i-i3-'93 

"            Pittsburg,  Pa.,  Penn.R.R.  .  .  b 

•(-1902 

255-0 

89.0 

N.  8-23~'o2 

Jersey  City,  N.  J.,  Penn.  R.R.  b 

1891 

252.7 

89.8 

B.  9~2o-'9i 

'  '           St.  Pancras  

1868 

240.0 

tl24.8 

74th  Regiment  Armory,  Buffalo,  N.  Y.  .  .  b 
Chicago  Coliseum  (old)  b 

ti1§6 

221.0 
215  .O 

94-o 
73-o 

N.  6-^-'oo 

Train-shed,  Cologne,  Prussia  a 

209.0 

78-7 

B.   10—  6—  '92 

Chicacro  Live  Stock  Pavilion                         a 

IOOC 

108  o 

B.  6—  28—  '06 

Dome,  West  Baden,  Ind  b 

Aywj 

195.0 

42.5 

B.  9—  4—  '02 

47th  Regiment  Armory,  Brooklyn,  N.  Y.  b 
69*         "              "        New  York  b 

11905 

189.8 

84.0 

N.  i2-23-'99 
B.  6-i-'o5 

Kansas  City  Coliseum  

187-3 

B.  7-5  -'oo 

Train-shed   Frankfort   Germany        .     .     b 

"f"i8oi 

184.0 

hnr    o 

M"       _                   » 

Dome,  Horticultural  Bldg.,  Col.  Ex.  

1892 

181.6 

>yj  -w 
91.0 

i^  .  y    LZ     y  I 

B.  '92-1-240 

St   Louis  Coliseum                                        b 

tiSoo 

178  c 

8O.O 

D        Q      Tf)—Jf\>7 

22d  Regiment  Armory,  New  York  

i  *°yy 

1889 

/°  -  o 

176.0 

t62.0 

97 

U    S   Gov  Bldg    St   Louis  Ex  b 

tI9°4 

172  .0 

66  8 

B    o—  20—  'o/i 

i  2th  Regiment  Armory,  New  York  

1888 

171-3 

55-6 

*  y  ^y    U4 

ist           "              "        Newark,  N.  J.  ..  b 

1900 

163.5 

73-3 

ST.  5-26-'oo 

ist           "              "        Chicago  

1894 

155-5 

77-5 

B.  '94-11-176 

Chicago  Coliseum  (new)                                b 

1802 

149.8 

T  2O    O 

66.0 

ST.  12—  24—  '92 

Machinery  Hall  Col   Ex                              b 

Dancing  Hall   Lattain  Beach                      b 

10y.£ 

1803 

•'O      '  w 

118.  7 

^       0 

"•   93~II—379 

1  3th  Regiment  Armory,  Scranton,  Pa  

±oVo 

112.  0 

49-5 

N.  8-24-'oi 

t  About. 


J  Clear. 


a  Two  hinges. 


b  Three  hinges. 


TABLES.  361 


KEY  TO   REFERENCES. 

A.  Civil  Engineers'  and  Architects'  Journal. 

B.  Engineering  News. 

C.  Weale's  Bridges. 

D.  Pennsylvania  Railway  Company's  Blues. 

E.  Wm.  H.  Brown,  Chief  Engineer,  Penna.  Ry.  Co. 

F.  Construction  des  Viaducts,  Tony  Fontenay. 

G.  Annales  des  Fonts  et  Chausses. 
H.  Mahan's  Civil  Engineering. 

I.  Masonry  Construction  by  Baker. 

J.  Spon's  Dictionary  of  Engineering. 

K.  Engineering. 

L.  Edinburgh  Encyclopaedia,  gth  Edition. 

M.  Scientific  American  Supplement. 

N.  Engineering  Record. 

O.  Engineering  Magazine. 

P.  Journal  of  the  Association  of  Engineering  Societies. 

Q.  Encyclopaedia  Britannica,  gth  Edkion. 

R.  Railway  and  Engineering  Journal. 

S.  Cresy's  Bridges. 

T.  Railway  Gazette. 

U.  Murray's  Handbook  of  Northern  Italy. 

V.  Le  Genie  Civil. 

W.  Messrs.  Keepers  &  Thacher. 

X.  The  Melan  Arch  Construction  Co. 

Y.  Transactions  of  Am.  Soc.  C.  E. 


INDEX. 


PACK 

Alexander  and  Thomson's  method 234 

Appendices 263 

Application  of  vertical  loads 159 

Applications,  Chapter  VII 159 

Arch-ring,  thickness  of,  at  skew-back 225 

Axial  stress,  effect  of 272,  283 

Brick  arch 228,  254 

Catenary,  equation  of 234 

"         two-nosed,  i  . 236 

"         transformed 235 

Circle,  the  three-point 238 

"      described 238 

Circular  arch,  ——-  =  constant: 

.A 

Curve,  general  equations  for 39,  88 

A<p,  general  expression  for 90 

Ax,  general  expression  for 93 

Ay,  general  expression  for 95 

Symmetrical  circular  arch  : 

Ac,  general  expression  for 97 

Al,  general  expression  for 96 

Symmetrical  circular  arch  with  two  hinges  : 

A(}>  (see  general  equation) 90 

Ax  (see  general  equation) 93 

Ay  (see  general  equation) 95 

Hi  for  horizontal  load,  Nx  included 42,  102 

"            "           "      "   neglected 41,  101 

"  changes  of  temperature 42,  103 

"         "         in  length  of  span 42,  103 

"   vertical  loads,  Nx  included 40,  loo 

363 


364  INDEX. 

PAGE 

ffi  for  vertical  loads,  Nx  neglected 39,  98 

V\    ' '  horizontal  loads,  Nx  included 42,  103 

"  "  "        "    neglected 42,102 

"  vertical  loads,  Nx  neglected 39,  99 

xa  for  horizontal  loads,  Nx  neglected 41,  102 

ya  for  vertical  loads,  Nx  included  40,  101 

"         "          "         "   neglected 40,  100 

Symmetrical  circular  arch  without  hinges  : 

A(p  (see  general  equation) 90 

Ax  (see  general  equation) 93 

v   Ay  (see  general  equation) 95 

HI   for  horizontal  loads,  Nx  neglected 44,   106 

"  changes  of  temperature,  TV7!  neglected 45,  107 

"  change  in  length  of  span,  Nx  neglected 107 

"  changes  in  Z/0o,  Nx  neglected 107 

"  vertical  loads,  Nx  neglected 43,   104 

general  expression  for 108 

M\    for  horizontal  loads,  Nx  neglected 44,   106 

"   changes  of  temperature,  Nx  neglected 108 

"   changes  in  length  of  span,  Nx  neglected 108 

"   vertical  loads,  Nx  neglected 43,  104 

general  expression  for 109 

V\    for  horizontal  loads,  Nx  neglected ". 45,  107 

"  vertical  loads,  Nx  neglected 44,  105 

ya,  y\,  and  yi,  values  of,  for  vertical  loads,  Nx  neglected 44,  105 

Comparison  of  four  types  of  arches  : 

If i   for  vertical  loads 144 

"   changes  of  temperature 153 

MX  for  horizontal  loads 155 

"    vertical  loads 151 

Mx  for  symmetrical   parabolic  arch  with  two   hinges,  vertical  loads 

only,  table  of  values 153 

Mx  for  arch  without  hinges,  table  of  values 152 

Fi  for  vertical  loads 146 

Vx  for  symmetrical   parabolic  arch  with  two   hinges,  vertical   loads 

only,  table  of  values 149 

Vx  for  arch  without  hinges,  table  of  values 148 

Stresses,  comparison  of,  for  three  types 156,   157 

Weights,  comparison  of,  for  three  types 155 

Comparison  of  results  of  tests  with  theory 254 

"        "      "    Douro  spandrel-braced  arch 221 

"        "      "   fixed  parabolic  arch 191,198,  201 

"  "        "      "   St.  Louis  arch 213,209 

"  "        "      "   Douro  bridge 186 

Concrete  arch 228 


INDEX, 


Concrete 254 

Conclusions  drawn  from  tests 255 

Co-ordinates  xa,  ya ,  etc 161 

Ax,  Ay,  As,  and  A<p,  general  formulas 6 

Data  for  St.  Louis  arch 204 

Deformation,  general  formulas  for 1 ,  6 

measurement  of 254 

general  formula  for  symmetrical  arch 50 

Diagram  for  Hi,  spandrel-braced  arch 221 

"         "    "    St.  Louis  arch. . .   208 

"         "  Mi,  St.  Louis  arch 215 

Distribution  of  loading,  masonry  arch 227 

"  "  pressure  upon  rectangular  section o 

"  "         "       general  formulas  for  8,  10 

Douro  bridge,  application  of  summation  formulas  to 182 

"  "        assumptions  of  loading  for 187 

"          "       relative  error  made  in  neglecting  Nx 189 

"          "       spandrel-braced 217 

Earth-filled  spandrels 230 

Equilibrium  polygon,  following  axis. 226 

"  "  equations  for  co-ordinates 17 

Examples: 

i°.  Parabolic  arch,  two  hinges,  vertical  loads , 162 

2°.   Parabolic  arch,  two  hinges,  horizontal  loads 164 

3°.   Parabolic  arch,  two  hinges,  temperature 165 

4°.   Parabolic  arch,  fixed,  vertical  loads 166 

5°.  Parabolic  arch,  fixed,  horizontal  loads 171 

6°.  Parabolic  arch,  fixed,  temperature 175 

7°.   Parabolic  arch,  fixed,  uniform  load 175 

8°,   Parabolic  arch,  fixed,  vertical  deflection 176 

9°.   Circular  arch,  two  hinges,  vertical  loads 177 

10°.  Circular  arch,  two  hinges,  horizontal  loads i 

n".  Circular  arch,  fixed,  vertical  loads.  „ 180 

Examples,  Alexander  and  Thomson's  masonry  arches 247-252 

External  forces,  general  relations  between, 14 

Floor-arches,  tests  of 258 

Formulas  for  practical  use 20-51 

Circular  arch,  hinged 39 

"          "      without  hinges 43 

Parabolic  arch,  hinged 20 

"  "       without  hinges 29 

Summation  formulas,  general 46—49 

Hi  (see  Parabolic  Circular,  etc.). 

Horizontal  loads,  point  of  application  of 160 

"  thrust  for  masonry  arches  242 


77 
73 

So 


366  INDEX. 


Integrals  employed  in  deducing  Ax 263 

Ay 209 

Joints  of  lead 229 

Keystone,  depth  of 256 

Lead,  joints  of 229 

Linear  arch,  Douro  bridge 182 

Loading  for  masonry  arch » 227 

Mi  (see  Parabolic  Circular,  etc.). 

Masonry  arch 223 

"       spandrels • 232 

Max  Am  Ende 217 

Maximum  stresses,  tabulation  method ....    161 

Merriman 258 

Monier  arch 254 

Moments  M\  and  Ma ,  character  of 162 

MX,  maximum  value  of   22 

Parabolic  arch,  £fJ  cos  <f>  =  constant : 

Curve,  general  equations  for ...  52,  53 

4<p,  general  expression  for  54 

Ax,  general  expression  for 55 

Ay,  general  expression  for 55 

Symmetrical  parabolic  arch  : 

Ac,  general  expression  for 58 

Al,  general  expression  for 57 

A$i,  general  expression  for 57 

Symmetrical  parabolic  arch  with  two  hinges  ' 

A(j>n    for  horizontal  loads,  Nx  neglected 64 

"  "  "      general 54 

"    vertical  loads,  Nx  neglected 60 

Ax   for  horizontal  loads,  Nx  neglected 64 

Ay   for  horizontal  loads,  JVX  neglected 65 

"   vertical  loads,  Nx  neglected 6 1 

Hi    for  horizontal  loads,  Nx  included 28,  65 

"          "  "         "  neglected 27,63 

"   temperature  changes 29,  66 

"   changes  in  length  of  span 29,  67 

"   uniform  loads,  Nx  neglected 23,  68 

"         "        load  over  all 24,68 

"   vertical  loads,  Nx  included 26,  61 

"         "          "         "   neglected 20,58 

Mx  for  uniform  loads,  Nx  neglected 24,  68 

"         "        load  over  all 69 

table  of  values  for  vertical  loads 153 

Vx  table  of  values  for  vertical  loads 149 

Vi   for  horizontal  loads,  Nx  included 66 


INDEX. 


367 


Fi  for  horizontal  loads,  Nx  neglected 27,  63 

"  uniform  loads,  Nx  neglected 23,  68 

"  "  load  over  all , 69 

"  vertical  loads 21,  58 

#0  for  horizontal  loads,  Nx  included 28 

"  "  "  neglected 27,63 

yo  for  vertical  loads,  Nx  included 27,  63 

"  "  "  ''  neglected 21,  59 

Symmetrical  parabolic  arch  without  hinges  : 

A<p   for  horizontal  loads,  Nx  neglected 80 

J(p0  for  vertical  loads,  Nx  neglected 73 

A(f>,  general  expression  for 54 

Ax  for  horizontal  loads,  Nx  neglected.  * 80 

"  vertical  loads,  Nx  neglected 74 

general  expression  for 55 

Ay  for  horizontal  loads,  Nx  neglected 81 

"  vertical  loads,  Nx  neglected 74 

general  expression  for 55 

HI  for  horizontal  loads,  Nx  included 35,  81 

"  "  "  "  neglected 33,77 

' '  changes  in  length  of  span 37,  84 

"  "  of  temperature 36,83 

"  "  in  0o ,  0/ ,  Ac,  etc 85 

"  uniform  loads 37,  86 

"  "  load  over  all  87 

"  vertical  loads,  Nx  included. 31,  75 

"  "  "  "  neglected 29,71 

Mi  for  horizontal  loads,  Nx  included 35,  82 

"  "  "  "  neglected 33,  78 

"  changes  of  temperature 37,  84 

"  "  in  0o,  Ac,  etc 86 

"  uniform  loads 37,  86 

"  vertical  loads,  Nx  neglected 30,  71 

Mt  for  horizontal  loads,  Nx  included 35,  82 

"  "  neglected 33,  78 

"  vertical  loads,  Nx  included 32,  76 

"  "  "  "  neglected 30,71 

MX  for  uniform  loads 86 

table  of  values  for  vertical  loads 152 

Vx  table  of  values  for  vertical  loads 148 

V\  for  horizontal  loads,  Nx  included 36,  83 

"  "  "  "  neglected 34,  78 

"  uniform  loads 37,  86 

"  vertical  loads,  Nx  included. ...  76 

"  neglected 30,72 


368  INDEX. 


x0    for  horizontal  loads,  Nx  neglected 34,  80 

Xi    for  horizontal  loads,  Nx  neglected 34,  79 

' '   vertical  loads,  Nx  neglected , 30,  73 

jra    for  horizontal  loads,  Nx  neglected 34,  79 

"    vertical  loads,  Nx  neglected 30,  73 

y«   y>  t  and_j»2  for  horizontal  loads 34,  79 

' '   vertical  loads 30,  72 

Symmetrical  parabolic  arch  with  one  hinge: 

HI   for  a  single  horizontal  load 140 

"   "     "       vertical  load 139 

M\   for  a  single  horizontal  load 140 

"   "     "       vertical  load 139 

V\    for  a  single  horizontal  load 140 

"  "     "       vertical  load 139 

Pins,  steel 230 

Reactions,  character  of 161 

"  for  Douro  bridge 187 

Resultant,  application  of 1 1 

"  "  "  for  several  forces 18 

Rough  quarry-stone  arch 253 

^-Semicircular  arch X^85-288" 

Seyrig .777  182 

Spandrel-braced  arch 218 

Spandrels  (see  Earth,  Masonry,  etc.). 

Special  formulas,  deduction  of 289 

Specifications,  masonry  arch,  Austrian 256 

St.  Louis  arch 204 

Stresses,  comparison  of,  three  types 156,   157 

Stress  diagram,  Douro  bridge 188 

Summation  formulas  applied  to  spandrel-braced  arch 217 

"         "  fixed  parabolic  arch 190 

"  "  "         "  two-hinged  arch 182 

Summation  formulas  : 

Symmetrical  arch  with  two  hinges  : 

Hi    for  a  single  horizontal  load 50,   130 

«'  ««      "      vertical  load 49,  130 

"   changes  of  temperature 50,   130 

Symmetrical  arch  without  hinges  ' 

Hi    for  a  single  horizontal  load 48,   129 

«   ««      «<      vertical  load 46,   129 

"  changes  of  temperature 49,  130 

Mi    for  a  single  horizontal  load 48,   129 

"   a      .«      vertical  load 47,   129 

Tests  of  arches 253 

Temperature,  St.  Louis  arch. .  _ 214 


INDEX.  369 

PAGE 

Tx,  maximum  value  of .  22 

Unsymmetrical  loading,  masonry  arches 245 

Variable  moment  of  inertia  : 

Symmetrical  arch  with  two  hinges; 

Hi    for  a  single  horizontal  load 50,  127 

<>  ..      «      vertical  load 49,  125 

"   change  of  temperature 50 

Symmetrical  arch  -without  hinges  : 

H\    for  any  symmetrical  loading 115 

"    a  single  horizontal  load 48,  118 

"  "      "      vertical  load 46,  117 

"   changes  of  temperature 49,  121 

Ali   for  any  loading 112 

"  a  single  horizontal  load 48,  120 

<•  <•      «      vertical  load 47,  119 

"   changes  of  temperature 49,  121 

Symmetrical  arch  -with  one  hinge  : 

Hl   for  a  single  horizontal  load 135 

"   vertical  loads 132 

MI   for  a  single  horizontal  load 138 

"    vertical  loads 134 

Fi   for  a  single  horizontal  load 137 

"   vertical  loads 134 

Vertical  loads  (see  Parabolic,  Circular,  etc.). 

"          "        replaced  by  force  and  couple 159 

"          "        point  of  application  of 159 

Weight,  comparison  of,  for  three  types  of  arches 155 

Wind  loads,  assumptions  concerning 160 

"        "      application  of •  i°° 


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19 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


APR  2  6    I954 
FEB  9       1955 


u-ui  i  '   ^r'^ 
JUL  2  8  1958 
JUL  2  9  teCP 
NOV  2  7  1961 
MOV  2  8REBI 

DEC  2  0  1961 


DEO 

WAR  8    1963 

MAR  5     RECD 

APRS    1963 

WAR  25 

XXl! 


2(A3105)444 


" 


Eigineering 
Library 


